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).

So, I couldn't sleep and decided to work on that. Here's the best I got until the moment - using only PHB.

I have done the math for 3 builds so far, all using Variant Human and fighting style Archery when multiclassed:

• Rogue 3, Crossbow Expert, Hand Crossbow +1;
• Rogue 2/Fighter 1, Sharpshooter, Heavy Crossbow +1;
• Rogue 2/Fighter 1, Crossbow Expert, Hand Crossbow +1;

Sneak Attack procs are assumed every round, as there are melee allies. I'm the first to say I was not very creative with the use of subclasses or magic items. As noted by András in the comments, Sneak Attack only procs once per turn, so the second Hand Crossbow attack does not deal the extra 2d6 damage, putting both builds roughly 2.5 average points below the Sharpshooter.

Assuming you you can hide to get unseen every turn

If Hide = Unseen = Attack with advantage is a thing (this depends on your stealth checks, but with expertise they should be very likely to work), the highest damaging build becomes Rogue 2/Fighter 1, Sharpshooter, Heavy Crossbow +1.

1. Cunning action for hiding.
2. Attack with advantage. Damage on hit = \$5.5 (1\textrm{d}10) + 3.5 (1\textrm{d}6) + 10 + 5 = 24\$. Damage on critical = \$11 + 7 + 10 + 5 = 33\$.
3. The probability of hitting without a crit is given by \$1 - (1 - \textrm{Usual probability})^2\$, where Usual probability is calculated using the +4 Modifier. Against ACs 16, 17, and 18, that works out to \$\{0.64, 0.5775, 0.51 \}\$ respectively.
4. The probability of a crit with advantage is \$1 - 0.95^2 = 0.0975\$.
5. The final average damage is \$ \{18.57, 17.07, 15.45 \} \$, for ACs 16, 17 and 18, respectively.

For the Rogue 3, Crossbow Expert build, assuming we can attack with the same hand crossbow twice, we have nine possible situations. Each one has a \$P_i\$ associated with it and an average damage \$ d_i \$. The average total damage is given by \$ \sum_{i=1}^{9} P_i \cdot d_i \$. Let \$ P = \{ 0.55, 0.5, 0.45 \} \$ be the probability of hitting an individual hit and \$ P_c = 0.05\$ the probability of landing a critical hit.

1. First attack hits, second misses. \$ P_1 = P \cdot (1 - P - P_c) \$; \$ d_1 = 3.5 + 7 + 5 = 15.5\$.
2. First attack misses, second hits. \$ P_2 = (1 - P - P_c) \cdot P \$; \$d_2 = 15.5\$.
3. First attack crits, second misses. \$ P_3 = P_c \cdot (1 - P - P_c) \$; \$d_3 = 7 + 14 + 5 = 26 \$.
4. First attack misses, second crits. \$ P_4 = (1 - P - P_c) \cdot P_c \$; \$d_4 = 26 \$.
5. First attack hits, second crits. \$ P_5 = P \cdot P_c \$; \$d_5 = 3.5 + 7 + 5 + 7 + 5 = 27.5 \$.
6. First attack crits, second hits. \$ P_6 = P_c \cdot P \$; \$d_6 = 7 + 14 + 5 + 3.5 + 5 = 34.5 \$.
7. Both attacks crit. \$P_7 = P_c \cdot P_c\$; \$d_7 = 7 + 14 + 5 + 7 + 5 = 38 \$.
8. Both attacks hit. \$ P_8 = P \cdot P \$; \$ d_8 = 3.5 + 7 + 5 + 3.5 + 5 = 24\$.
9. Both attacks miss. \$P_9 = (1 - P - P_c) \cdot (1 - P - P_c)\$; \$d_9 = 0\$.

The result of the summation, if I did everything right, gives us an average damage of \$ \{16.92, 15.79, 14.62 \} \$.

Similarly, for Rogue 2/Fighter 1, removing the 1d6 from Sneak Attack, we get \$\mathbf{d} = \{12, 12, 19, 19, 24, 27.5, 31, 20.5\} \$, but increasing the Probabilities to \$ \{ 0.65, 0.6, 0.55\}\$, we get an average damage of \$ \{ 15.66, 14.70, 13.73 \} \$.