Other than having a low item and level dependency (as mentioned here) a good roamer needs to be able to do a few things:
- Counter-jungle
- Deny the opponent's solo
- Gank
I'll talk about why each of these are important, and what champions other than Alistar fit these roles very well.
Counter-Jungling
Counter-jungling is simply screwing with the enemy jungler. Junglers are countered by stealing their buffs and denying respawn.
Stealing buffs can be crippling to many junglers, as buffs are needed for clearing the jungle faster and ganking. For example, a jungler without crowd control needs Lizard to slow when ganking. A jungler who needs their abilities to jungle effectively (Nunu/Shaco) relies heavily on Golem. Stealing either of these will not only deny the jungler experience but likely prevent them from ganking other lanes, making your allies lives easier.
Denying respawn is simply done by killing the highest experience value creep in a camp of the enemies jungle. This gives less experience to the enemy jungler and the creeps will not respawn until the jungler kills the entire camp. Good creeps to do this with are the Big Banshee, Big Wolf, Lizard, and Golem.
Keep in mind that going for enemy buffs is extremely risky; most teams will actively protect their jungler and you might mind end up in a 1v3. A roamer needs to be able to deal with this.
Denying Solo
Simply put, prevent your opponent's solos from getting experience. Running into a lane and harassing the solo enough that they have to blue pill back home is basically all that you need to do. This gives your team's solos some free time to level and get last hits without harassment.
Ganking
This largely happens along with denying solo. In order to do this a roamer needs crowd control so that your allies can get in to do damage. Even if your gank fails, as long as you manage to get the enemy champion to blue pill back home you have successfully given your allies an opportunity to freely farm their lanes.
An important thing to remember is that you do not want to get the kill from a gank, unless the target will escape without you taking the kill. It is more important for your carries to get the gold in order to carry your team late game.
The glue that keeps it together
To accomplish the above goals you need to move quickly. This usually depends on the boots you choose, but getting the gold for those boots can be difficult as a roamer, so inherent speed is important.
The less time you spend getting from lane to lane the less experience you will lose while traveling. Although this isn't a huge deal, because you losing experience means your allies are gaining more, you still need to be able to travel quickly in order to take advantage of opportunities when they present themselves. Fast traveling means you can run to the enemy jungler when you see him at the buff you warded, run to a lane to gank, or escape from a nasty scenario where the enemy team decides they don't like you roaming.
Alternatives to Alistar
Addressing all of the above roles, I have a few champions suggestion other than Alistar.
As stated in other answers, a champion who can jungle tends to make a good roamer, because both roamers and junglers have similar roles (deny solo, gank, counter-jungle).
Nunu in particular is a great roamer because of his abilities. Consume is a natural Smite, ideal for stealing buffs. My games usually start with me warding the enemy Lizard or Golem (depending on the enemy jungler), and waiting until they bring the creep down low so I can flash in and Consume. If that proves too risky I waltz to the banshees and consume Big Banshee for an easy respawn denial. Blood Boil is a great way to get around the map quickly, especially with Boots of Mobility. Less time traveling means you lose less experience and get more opportunities to be present for a gank or push. You can also cast on an ally to make them even deadlier during a gank while still getting the advantage for yourself. Ice Blast is a wicked ganking tool, providing a massive slow that is a nice lead into the ultimate Absolute Zero. The combo should give your allies plenty of time to run in and take out the target, but be careful not to take the kill with Absolute Zero's high damage.
Evelynn also makes a good roamer. Her ganking is ridiculous, and I won't bother going over how it works. Instead of stealing buffs, unless you have Smite, she tends to be more effective at scaring off junglers when they try to take a buff. Especially at level 1, many junglers get very low hp while taking Golem or Lizard. It's an ideal time to kill them with a stun/Ignite, or at least scare them off. If you take an Ignite/Smite combo you might get both a kill and a buff (win!). Going invisible makes Evelynn impossible to catch early game unless someone invests in an invisible detection wards (which will not happen), so there's little risk to counter-jungling. Her ultimate provides a massive speed boost around mid game, making her a very fast traveler, especially with Boots of Mobility.
There are plenty more roamers that I won't care to mention but given the criteria it should be easy to point out which of your favorite champions can make a good roamer.
In general, I would say it is an absolutely horrible idea to not have smite.
Smite allows you to clear the early jungle faster, but its more important aspect is control over Dragon and Baron. Most junglers will keep smite up as much as possible in the late game for Baron and Dragon so they have the ability to steal or take the objective more easily.
A stolen Baron or Dragon can easily help sway a game in a team's favor. The global gold (and buff in the case of Baron) is a huge boon to a team. It can allow you to stay in game if you are behind or seal the deal if you are ahead. By not running smite you are putting that at risk.
You might be able to get away with no smite if the enemy also lacked it, but you are making the taking of Dragon and Baron that much more of a gamble. If your team has smite and the enemy does not, then it is infinitely easier to steal both those objectives.
Nunu is one of the few junglers that could run without smite. His consume does comparable true damage to neutral creeps allowing for a lot more objective control compared to junglers without smite. Cho'Gath has a similar ability with his ultimate Feast, since it does such a large amount of true damage to neutral monsters.
However I'd still run Smite, it is just that important.
Update:
With Season 3¹²³, Riot has moved the jungle closer to its season one state wherein the monsters are much stronger. To compensate they have included some jungle only items (Hunter's Machete, Spirit Stone, etc) but overall the difficulty has been ramped up. This makes smite much more important for no/low sustain junglers. Without using smite you have to suffer much more damage from the larger camps, forcing you to fall behind as you retreat to the fountain to heal.
Best Answer
At the most basic level (to avoid not attacking)- you don't need to click continuously; just right click once and your champion will attack until you tell them to do something else.
Eventually (and this really isn't important now, but I include it to not lead you astray in the long run) you'll find that this is slightly sub-optimal when chasing, because your champion will only walk forward when actually out of range, which is problem when you're not out of range but soon will be. In this situation, you should manually walk forward after each attack. You'll need familiarity with the champion in question to know exactly how early in the attack animation you can move without canceling it.