Note: This was written prior to patch 1.05. With the changes to defensive skills such as Leap and War Cry, and in general making Inferno easier, some of this advice needs reevaluation. Novarg left a solid answer that I can't confirm since I don't play anymore, but seems to address the changes in later patches that allow for a more damage-oriented build.
Strength/Vitality gear, though it got you through Normal, Nightmare, and probably even Hell just fine, won't cut it anymore. You're going to need armor and resistances. And a defense-oriented spec. The advice below is geared towards Act 1, though with minor tweaks it holds up pretty well in Act 2 and Acts 3 & 4 (see bottom of post). The Barbarian is a very gear-oriented class, and our defensive skills can make up for a lot of shortcomings in gear. As your gear progresses, you'll find that you can make your build a little more offense-oriented, so don't be afraid to experiment and swap in some more powerful offensive abilities if you find that your survivability is doing fine!
Skills
I've seen a few different builds bandied about, but most will look pretty similar to what I've got below, other than maybe a rune or a passive different. Here's what I'm using. I'll try to explain why I've chosen what I did, and what other options I've seen.
Frenzy gives high single-target DPS, which you'll need. I'm low on Life on Hit gear but have a decent DPS weapon for Act 1, so I've got it runed with Triumph so that I have a fairly steady stream of life coming in (6% of my max health) and can save money on potions. If your weapon is kind of weak or you just don't have a problem topping off your health, a far more popular option is Maniac for the +20% damage.
Sidearm also adds nice damage but I don't like the unpredictability of it, as sometimes it shoots off and hits a treasure goblin I didn't want to deal with yet. There's also Smite for the 20% stun (which doesn't last as long on elites) which helps you out a bit with reducing damage, and then there's Vanguard for the movement speed bonus. It's very hard to go wrong with runing this and it pretty much comes down to a matter of personal preference or filling gaps in your gear. I switch this around quite often.
If I had to pick one, I'd probably settle on Maniac. Even though as far as just the Frenzy attack is concerned, Sidearm does better damage, one nice benefit of Maniac is that though the attack speed bonus from Frenzy doesn't transfer to other skills, the +20% damage bonus does. This means your Revenge is hitting harder. Speaking of which...
I haven't seen a single general-purpose Inferno build that skips Revenge, and that's because this is the skill that keeps us alive. Does nice AOE damage to the enemy and heals us 5% of our max life for every enemy hit. This is the best tool in our toolkit. Unfortunately, it can be a little unreliable sometimes, which is why runing it with Provocation to get the trigger chance up to 30% is so popular. This skill is a must-have. Not to mention, the simple damage of it can be pretty impressive.
There are other skills you could replace this with, but Leap runed with Iron Impact for the +300% armor is a great defensive ability. It means for 4 seconds of every 10, your armor is much, much higher. It's also useful for getting yourself away from scary elite packs, walls, arcane beams, jail, etc. Just make sure you're constantly leaping... you want that +300% armor up as much as possible.
I've seen builds that replace this with Ground Stomp for the stun or Overpower runed with Revel for the heal, but I like Leap a little better simply for the utility of getting out of a hot spot or taking shortcuts in dungeons.
Ignore Pain is your "Oh Crap!" button. Reducing damage by 65% can be especially powerful if combined with the 4 seconds of +300% armor during Leap. I find that I use this less as a "run away!" button and more as "do or die!" button, usually popping it right after I've decided I'm going to go toe-to-toe with a big nasty after popping Wrath of the Berserker. It gives me 5 mostly-worry-free seconds during which I can try to burn the mob down. Throughout Act 1 I runed it with Ignorance is Bliss so that I was getting 20% life back, but once I got to Act 2 I switched to Iron Hide because I found I needed those extra seconds of duration.
As a bonus, you can pop Ignore Pain even while frozen! Needless to say, this can be a real life saver.
Wrath of the Berserker runed with Insanity for +100% damage is a beast. With a 2 minute cooldown, this is what you pop when you run into a scary situation or see an elite that you want to burn down quickly. Combine with Leap and Ignore Pain to be as close to invulnerable as you're ever going to get for about 5 seconds. Sadly, not as close as you'd like, but you'll still be quite a menace! Make sure you smack those revenge procs as much as possible during the 15 seconds of this skill, because with the +100% damage from this, the +20% from a full stack of Frenzy, and the +25% from Berserker rage, you'll be able to pile some serious hurt on. In Act 1, popping this was often enough to take down at least one member of a pack, if not the whole pack. In Act 2... well, I counted myself lucky if it let me burn one pack member down.
One thing that really can't be understated about this skill is that you become immune to crowd control effects (Fear, Freeze, Jailer, etc) while this buff is active. It makes it great for fighting the elite packs that are traditionally a big problem for melee.
War Cry will give you more armor, and if you rune it with Impurity, it'll bump up your resistances up by 50%! As far as I can tell, this skill and rune are pretty much required for Inferno, as they're needed to get your armor and especially resistances up to the needed levels. Don't forget to refresh this every minute... you do not want to let it expire mid-fight.
Passives
Due to the lack of fury spenders in my build (except for Wrath of the Berserker, which is on a 2 minute cooldown), I go with Berserker Rage for +25% damage when at Max Fury, which is most of the time. Even if you start a fight at 0 fury, a War Cry + Leap will fill 45 of your 100 for you, and your Frenzied attacks and inevitable Revenge proc will fill the rest up quickly.
As I got to Act 2 and had problems with survivability, I ended up having to spend passive elsewhere. I've floated between Superstition for the damage reduction and Pound of Flesh for the more frequent and powerful health orb drops. I think this really comes down to a matter of preference and gearing in Act 2... you'll use it to shore up your weak spots.
For the other passives, I took Nerves of Steel and Tough as Nails for the armor bonuses. Swapping one out for Superstition also seems popular, though I've read a lot of arguments about it, with one popular view being that Superstition doesn't pay off as well since the bulk of your damage is physical, physical tends to hit harder, and many magic attacks can be dodged. Find what works for you.
Gear
Going from Hell to Inferno is about the same difficulty jump as going from Act 1 Inferno to Act 2. Unless you've got a wealthy benefactor, you're probably going to have to spend some time farming Act IV Hell to either find items or raise the cash to buy some decent items of the AH. Unless you were incredibly lucky with drops, you can't expect to just stroll into Inferno the second you beat Act IV Hell and start kicking butts and taking names. As you buy gear on the Auction House, keep an eye on the fact that you'll be wanting to move on to Act 2 pretty quickly. If you can get +20 resist all armor for 100k but +60 resist all armor for 300k, save a little longer and buy those 300k gloves. You'll need that extra resist when it comes time to do Act 2, and it'll help make your Act 1 Butcher runs easier too.
It goes without saying, you still want Strength and Vitality when you can get it
Hit points and damage. Yes, please. And Strength counts as armor, which is a bonus. You still want these where you can get them, but the simple fact is that you'll probably have to sacrifice them a bit to get your armor and resistances up to the needed level. Prioritize Vitality over Strength.
Find a balance between Armor and Resistance
This forum post has a pretty good description of how armor and resistances interact. Seriously, go read it now. I also took a stab at summarizing how Damage Reduction works in an answer here. Understanding how armor and resistance work is going to be key to your gearing strategy. Basically, you need to figure out where you're at on this chart so that you can figure out how much armor a single point of resistance is worth to you, so that you can min-max your gear. If you don't want to look through all that math, it amounts to this... you need a good balance of armor and resistance. You don't want to just go all-out in one or the other, because you get the best returns by keeping them at an equilibrium.
As far as Intelligence goes, 10 points of Intelligence = 1 point of resist all, so while you don't want to specifically focus on Intelligence, you can at least look at it and mentally convert it into Resist All when looking at gear. Just pretend that piece of gear that you're looking at with 38 Resist All and 130 Intelligence is really 51 Resist All. I wouldn't specifically go searching the AH for high intelligence items though unless you're really struggling to come up with enough resistance. It's just not a terribly efficient way of getting your resistances up.
A good target for Act 1 Inferno is 600 in resistances and 7-8k in armor (buffed). Yes, this means you'll probably need to hit the Auction Hall. You can get by with less... don't think that you have to wait to play Inferno until you've reached these goals. I was getting by just fine with about 7k in armor, 480 in resists and no shield (but 16k DPS) if I played solo. As you add players, though, those enemies 75% HP buff for each additional player, which means you'll start feeling a lot more pain from weak gear. It's also tricky as you add players because one big tool you have is Wrath of the Berserker to burn elite mobs down quickly before they can do you a ton of harm, but you have no chance of burning down an elite with 430% it's normal health by yourself. Your team has to be very well coordinated and I've found it's often important for them to kite some mobs from the pack away.
It's a little bit more of a moving target in terms of what you need to be successful at Act 2. I'm currently at about 900 resists and 9000 armor (buffed) but only 12k DPS, and I'm still dying quite a bit, but I can grind my way through Act 2, taking a small number (~10%) of elite packs without a death, a few more (~30%) with 2-3 deaths, and that means the rest (~60%) I haven't had much luck taking out. Given the rarity and cost of the resist gear, you'll probably find yourself faced with a choice between going for resists or going for damage. I've seen videos of people who are quite successful in Act 2 who are only at about 500 resists and 7k armor, but are up around 35k DPS. A lot of this will come down to what gear you happen to find or afford.
No, really, focus on armor and resistance.
I'm repeating myself here, but armor and resistance is king. Having a huge health pool alone is worthless. You're not going to want to throw away +100 vitality for a mere 100 armor, but you probably will want to throw away 30 vitality for 200 armor and 30 resist all. One thing to remember, though, is that if you've got Nerves of Steel, Vitality is Armor for you. Don't get me wrong, you don't want to abandon vitality, but you'll probably have to trade some off to get the armor/resistance you need. I had about 40k health running through Hell. I'm early in my gearing process for Inferno, but I'm now down to about 30k but with much higher armor/resistance.
Once you hit Act 2, you'll want to try to stack more Vitality, probably around 45k, so that you can better survive the damage spikes.
Drop some cash on a high-DPS weapon
Note: Take these numbers with a grain of salt. Higher-DPS weapons were harder to come by back when the game was new, whereas 900 DPS one-handers now go for a pittance. But the theory holds... buy the best DPS one-hander you can afford.
If you can swing it, you'd really like to have a 800+ DPS one-handed weapon. This should be your first purchase when you hit level 60, because it'll make farming everything else even on Hell difficulty a lot faster. Walking into Inferno with that 350 DPS weapon that carried you through Hell difficulty just isn't going to cut it, and you also want high damage so you can dish out some good AOE on your Revenge procs.
Your ultimate goal is to get a 800-900 DPS one-hander (obviously, even higher would be better!) so that you can sword & board, but you may not be so lucky as to find one or be able to afford one early. There are, however, often some very cheap high-DPS 2-handers for sale on the AH, and using that until you are able to replace will allow you to start running Inferno early instead of being stuck farming in Hell.
Seriously, don't be afraid to use a blue weapon with high DPS as a stepping stone. Upon hitting 60 in Act III of Hell, I immediately went out and bought an 800 DPS 2-hander for 200k on the AH. My overall DPS doubled. Even though the weapon I got is just a stepping stone to the one I'd eventually like to have, it was worth not sitting there and waiting until I had a few million gold.
When you hit Act 2, you'll feel that same pain all over again, and you'll find yourself now wanting a 1200 DPS one-hander. I haven't been able to afford one yet, but I bought a 1200 DPS two-hander that I pull out specifically for the 15 seconds I've got for Wrath of the Berserker so that I can make the most of that time. Once my time is up, I switch back to sword & board. Damage output simply can't be ignored in Act 2, because unless you've got millions to spend on the AH, your armor simply won't be good enough to let you keep soaking damage from elite packs for 3-4 minutes straight.
Those Life Steal properties that used to be so awesome? They suck now. Look for +Life on Hit instead
Life Steal (where you gain life equal to some percent of the damage you dealt), which I had found completely awesome through Normal and Nightmare and still pretty decent even on Hell, is pretty worthless on Inferno, as it scales down at that difficulty level. I haven't run the math myself but I've seen the number 20% thrown around a lot, so I'm going to assume its correct. That means if you had 10% worth of life steal gear (which is pretty hard to achieve, since I think you can only get it on Weapons, Belts, and a passive skill), that would amount to only 2% on Inferno. AOE attacks normalize it, so it's not like you will be raking it in that way. Plus, if you use the build above, your only AOE attacks are Leap (which is weak and every 10 seconds) and Revenge (which already does its own healing). Let's say you're cranking out 15k DPS... that's going to amount to about 300 HPS. Not great.
However, +Life on Hit doesn't appear scale down on Inferno, so this is what you want to get if possible. I've seen over 600 Life on Hit on weapons.
Act 2 and Beyond
I haven't beaten Inferno yet, mainly due to time constraints, but I have made it through to Act III and about halfway through Act IV, and can kill about 75% of the elite packs I encounter in Act III and IV. I do it with a build pretty similar to the one I just described for Act I, except that in Act 2 I had to drop Berserker Rage and take Relentless instead. As you get better gear and a higher health pool, Relentless becomes very useful since against elite packs you'll often be in that <20% health range, and the 50% damage reduction makes a huge difference. I also find on Act 3 that I need both other passives to be Nerves of Steel and Tough as Nails for the armor bonuses, plus the Enchantress as my ally for even more armor bonus. Eventually gear may overcome this, but you definitely need to be very defense oriented in Act 3. I have yet to find a great weapon, so I'm doing this all with only about 10k DPS. It's slow going, but it's survivable.
Eventually, you'll get earlier acts on farm mode (yes, even Act 2), and will be able to change your build to be more offense-oriented so you can more quickly farm those acts. My gear has progressed to the point on Act 2 that I can run with Inspiring Presence, Berserker Rage, and Battle Rage runed with Marauder's Rage for the +30% damage bonus. Experiment with this as your gear progresses, because trading off survivability that you've got more than enough of is worth the +55% DPS boost.
As you gear up, you'll also definitely need to get in the habit of just skipping some of the elite packs that are too tough for you. You'll be wasting precious time on your Nephalem Valor buff dying repeatedly on them. Just train them off somewhere out of the way and die so that you don't have to worry about them anymore. Then move on to the packs that you can handle.
As far as farming, when you first get to Act II I found Oasis to have the most Barbarian-friendly mobs, and as a bonus it sometimes spawns dungeons with Resplendent Chests, or even better, the Treasure Vault of Khan Dakab. Eventually you will be able to facetank every elite pack in Act 2 with this spec... it's just a matter of getting your vitality, resists, and armor up.
Also, make sure you check out yx.'s answer about advice on how to gear up for the later acts.
Best Answer
I find the first quest of Act 1 on Inferno is a good place to farm gold after the nerfs to the other hot spots recently, essentially start from the 'Rumford at the gate' quest, and clear the area outside of New Tristram followed by clearing Old Tristram, killing the 3 Wretched Mothers and the Queen, and then returning the quest and starting all over again.
During this clear you have the possibility to find:
As a general rule, I go for more clears = more gold, so will skip any cellar with an elite in them. On a poor run (neither of the wells spawn and you don't get the treasure horder) you will make about 10k excluding selling gems that drop.
The wells are worth 10-20k each depending on your gold find.
The dank cellar is worth 10-50k depending on the current cost of gems on the auction house.
Each clear should take no more than 5 minutes.