[RPG] Optimized DPR at 10th to 13th level for a Vengeance Paladin

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).

If you're looking to boost damage, the best way to do it is by boosting damage. To address your specific choices:

• Alert will slightly boost your damage by letting you deal damage when you would normally be surprised.
• Charger is mediocre at best before level 5, (which is one level away!), and just plain sucks afterwards. You have better things to do with your action and with your bonus action.
• Heavy Armor Master will boost your damage significantly, since it will increase your Strength score to 16, giving you +1 to attack rolls and damage rolls. It will also help you survive, which, in the long run, will increase your damage a bit.

As you've probably noticed, there is not a lot of feat support for sword-and-board warriors. Most of the damage boosting options go to 2-handed builds. One option you didn't mention is Shield Master. It allows you to knock foes prone with your bonus action, which can help your attacks hit, thereby boosting your overall damage. But, to be perfectly honest, you're far better off getting your Strength score up to 16.

Given that getting your Strength score up to 16 is the best boost to your damage that you can give yourself right now, it's definitely a priority. So I would recommend either Heavy Armor Master, or just taking a stat increase and getting your Charisma to 16 as well as your Strength.

"Wait, what? How does boosting Charisma boost my damage?"

Well, it doesn't directly boost your damage. But neither does Heavy Armor Master. And while in the short-term, Heavy Armor Master will make you a lot tougher, as the game goes on 3 damage will become less and less significant, and more and more monsters will bypass Heavy Armor Master anyway.

On the other hand, Charisma helps with almost everything else you do - your spells, your Channel Divinity, and, from level 6 onward, all of your saving throws and all of the saving throws of any ally standing near you. Where Heavy Armor Master will get less and less important, this will get more and more important.

So, final recommendation: Either take Heavy Armor Master, or take the ability score increase and get both Strength and Charisma up to 16. They're both good options, and they'll both boost your damage roughly equally. The main difference is that Heavy Armor Master is much better in the short-term, and Charisma is much better in the long-term.