Prioritize number of attacks and Feat combinations
Ok, a quick addendum before we go any further. There is no such thing as one perfect build for every situation. The environment you'll be fighting in, the type and number of opponents, and even the nature of the story you're engaged in can all change which characters are "best" suited. For example, a character who hits once but hits extremely hard (like a rogue) may have higher DPR (damage per round) than another, but against a horde of weak enemies (e.g. Goblins) they may find themselves less useful than their DPR would suggest they would be, since a large amount of their damage each round is wasted hurting an already dead enemy.
All that being said, I've done a bit of research and found that at this particular level, there is a build that will work very well in a lot of scenarios, and meets the criteria you've set (survivability and DPR without any other players' help) in a measurable way. So enough hedging: let's look at the details.
The Build
Stout Halfling (counterintuitive, but hear me out), Eldritch Knight Fighter 13, Rogue (or Wizard) 2
Str: 8
Dex: 20
Con: 16
Int: 14
Wis: 12
Cha: 8
AC: 18, 20 with Haste, up to 25 with cover
Feats: Sharpshooter, Crossbow Expert
Favored weapon: Hand crossbow
Favored Spells (in order of importance): Haste, Mage Armor, Shield, Find Familiar (or another 1st level Wizard spell), 7 more evocation or abjuration wizard spells of your choice (you'll rarely be casting them).
Rationale
In descending order of importance, the three factors you need for maximum ranged weapon damage dealing are:
- Multiple attacks
- Accuracy & Damage Bonus (about equal: both come from Dex)
- Base Damage (weapon's dice)
Multiple attacks are almost always the most important feature. A typical enemy in DnD will have an AC which can be hit on a natural 8 or better. So usually, an increase an increase in accuracy (say, the 2 point increase from the fighter's "archery" style) will grant you about a 16.666% overall boost in expected damage (hitting on 14/20 attacks rather than 12/20), while a boost in number of attacks will grant you at least a 20% boost in damage (going from 5 attacks to 6, which usually can only happen at 20th level anyway). Although enemies with particularly high AC may alter this math, on average multiple attacks will be of more benefit. That being said, we're going to get both as high as possible.
With all that in mind, you're going to want the following build:
- 13 levels in Eldritch Knight: 11 for 3 standard attacks. At level 13, trade in the spell you gained at 8th level for "Haste", and get another single weapon attack (4 per round total so far).
- At level 4, take crossbow expert feat. Now, according to sage advice (http://dnd.wizards.com/articles/features/sageadvice_feats), you will be able to use your bonus action for another hand crossbow attack. Although the hand crossbow has the lowest damage dice of the martial ranged weapons, this feat makes it the most damaging in most circumstances. (5 attacks per round by level 15).
- At level 6, take sharpshooter. Now, you can shoot your hand crossbow just as well up to 120 feet (and since most combat in a dungeon crawl happens within 100 feet, you're golden). Also, as long as your target's AC is less than 22, you always want to take the -5 to roll for +10 to damage (the average damage per round will be higher).
Your remaining two attribute increases should got to getting a 20 dexterity. So far we're up to: 5 attacks per round, expending nothing other than a 3rd level spell slot (2 per day available) and ammunition. Each attack does 1d6+15 damage, or 18.5 average damage per successful attack. That's (averaging out the rolls) a whopping 92.5 damage per round, without even using action surge! Even more amazingly, you could still be dealing an average 37 damage on a turn in which you are DODGING (haste action + bonus action attacks)!
Even without haste, we still could be doing an average of 74 damage per round (assuming all attacks hit) with the hand crossbow, expending no resources other than bolts.
For the final two levels, there are a couple of options; but probably the best one is to take two levels of Rogue. You'll get an extra 1d6 per round from sneak attack, and cunning action (which, although it would lose one weapon attack, would allow you to disengage, dash, or hide as a bonus action: it'll give you considerable versatility on the battle field, and allow you to easily keep your distance). Also, you could get expertise in stealth and acrobatics, which will be a huge boon if you're hoping to scout or hide in combat, or avoid being grappled. Our damage total (without action surge) is now up to 96 damage per round. If you wanted to Haste more often rather than do a bit more damage (and losing some battlefield mobility), you could take 2 levels in Wizard instead, allowing you 3 castings of haste automatically, and an Arcane Tradition (War Magic could fit nicely here, or Bladesinger if your DM allows it for a non-elven character. Warning: a DM may consider an attack with a hand crossbow to be using "two hands to attack with a weapon", which ends the bladesong. Check with them before choosing a subclass).
The Halfling's bonus to dexterity is essential, but you could get this from other races. A real benefit is the +1 bonus to constitution. It may not look like much, but this build is very much about maintaining the Haste, and has no attribute boosts to spare. If you take this build, DC 10 concentration saves (which is what most attacks will incur if they hit you; only attacks dealing 22 or more damage will up this difficulty) will only fail on a natural 1: and because of Halfling's "Lucky" trait, this means the standard concentration save will only fail 1 in 400 times. Also, the Halfling ability to move through the space of medium hostile creatures makes this build insanely mobile. If you went with 2 levels of Rogue, even if your character was completely surrounded by (medium sized) guards, you could Disengage (cunning action), Dash 45 feet (Haste dash action, speed doubled, five feet of difficult terrain for moving through an enemy's space), make three attacks (action), then move up to 50 feet more, all without incurring any opportunity attacks.
For your non-abjuration/evocation first level spell gained at level 3, I'd suggest Find Familiar (an Owl using the Help action in combat can be very useful). You should pick up Mage Armor right away at level 3. With mage armor and Haste activated, you'll have an AC of 20 (13+5 from dex+2 from Haste), before you even take cover (which you should be doing as much as possible as a ranged character). This will make your standard AC between 22 and 25. If you pick up Shield at level 5, you can even bring this up to 30 for a round, in REACTION to being hit (which could cancel the hit in question) at the expense of a first level spell slot.
There are other options that make for powerful ranged damage dealers at this level (Level 10 bard with swift quiver and heavy crossbows will fire 4 times per round, and ranger5/rogue* beats out fighter at most levels). But this build has a slight (or major) numerical advantage over all of them I've been able to calculate at precisely level 15. One last consideration: if you went Fighter 14, Rogue 1, you'd get the same max damage, lose the cunning action, and get access to one 3rd, 2nd, or 1st level Wizard spell of your choice. Mirror image alone might be worth this: but it all depends on how much flexibility you want on your turn. Personally, I'd go Rogue 2 (cunning action is fun and useful).
Best Answer
There are two basic strategies you should be employing to survive your early, "squishy" levels, and luckily they're both available to you at level 2:
1. Avoid taking damage
a. Make attacks from range: most enemies you face will find themselves having to choose between trying to take out the ranged attacker (you) in the back vs. the melee combatant right in their face, and will choose the melee combatant. Being at range generally decreases your frequency of being targeted.
b. while you're at range, make use of cover: after taking your shot you should be able to duck back behind something, giving you a bonus to AC. (See PHB p.196 for details; note that even having an interposed ally should bump your AC--many tables forget about this.)
c. use your cunning actions: at level 2 you've already got the ability to Dash, Disengage, or Hide in addition to attacking. There's almost no round of combat where one of those can't help you avoid being harmed.
(You are foregoing fighting with two weapons by emphasizing cunning actions, as either choice consumes your one bonus action. But that's alright, because you'd need to be up in melee to make good use of two weapons, and you (currently) don't want to be there!)
2. Rely on your companions
And sometimes you're going to get knocked down. Hopefully you're getting sneak attack on your ranged attacks so doing 1d6(shortbow)+1d6(Sneak Attack)+DEX from range. This is really good damage to deal at level 2: your companions should see you as a valuable contributor during combat. Worth healing, perhaps. (I mean, every one of your companions could choose cure wounds at this point; hopefully someone has. And the paladin has 10HP in laying of hands....)
But even if nobody wants to spend actions and spells slots healing while in combat, it's certainly worth your paladin and wild-shaped druid friends soaking damage and holding a front line to allow you to do your thing.
But you shouldn't just assume this is the case: discuss with your party your current need to stay free(er) from harm, and make sure the the front-liners don't mind taking the majority of damage and appreciate your contribution.
(While you're at it, make sure you coordinate tactics with them: making sure that you have an enemy to target with an ally adjacent almost doubles your damage, thanks to sneak attack!.)
Coda: fix the problem
I'd be remiss if I didn't point out that you'll do better on average, in the long run, taking the fixed HP value at level-up than by rolling. (See Why would I ever choose rolling hit points?.) Even if you like the excitement of rolling, the impact of increased variability when you're at a low level can be... problematic. As you've learned =)
Encore: it gets better
The first few levels of 5e are notoriously dangerous for PCs. Two hits can easily take anyone down at L1, and there's precious-little healing in a L1 party. Spare the dying and drag off the bodies is the best you can do some days.
Remember that these first levels ("tier 1") are considered "apprentice" levels (PHB p.15). Your character's not really expected to be great yet, they're developing the skills and habits of greatness. You rolled badly, so your tenuous L1 experience is stretching into L2. This, too, shall pass, and quickly it will pass. And just think about all the cool things you'll get at level 3: an archetype, a second sneak attack die, party-mates with second-level spells... it's going to be awesome!
(As one commenter points out, at L4 if you want to stay a ranged-rogue, you may want to consider the Crossbow Expert feat. Or you may want to start making your way into melee, being a stabby-stabby rogue. Either one can work very well, and I don't think you've yet closed off either option.)