The short answer:
AC 0-5: Human Great Weapon Fighter with the Great Weapon Master feat wielding a Greatsword
AC 6-20: Raging Human Barbarian with the Polearm Master feat wielding a Polearm
AC 21-26: Half-Orc Two-Weapon Fighter
AC 27+: Human Great Weapon Fighter with the Great Weapon Master feat wielding a Greatsword, part II: the GWF returns.
The significantly longer answer is that here are some calculations I did for the options that seemed obvious to me. Interestingly enough, the Barbarian didn't make the top of the list in terms of pure damage, but it did make the top of the list once we added AC into the mix.
You can also add Caltrops to any of these builds (and you can afford a lot of caltrops) to add up to 1 dmg/r.
Other notes:
- The only source material used for this was the PHB, as I do not have access to all of the adventure paths and supplements
- Magic users need not apply: Cantrips are almost universally low-damage, and they never get to benefit from base stats to get those sweet, sweet flat bonus damage numbers. All other spells are too consumable to be considered, so they didn't make it to the number crunching stage.
- Rogues were briefly considered, but sneak attack just isn't reliable enough when you can't count on nearby allies or advantage.
- No cheese! I tried to stick to a purist approach here. There may be strategies that your GM approves like this but most reasonably GMs would ban that kind of build from their tables for being too OP. This list fits nicely within very vanilla RAW and RAI.
- I gave all characters rolls of 18 for their stats, then added any racial strength bonus that applied. Point-buy only allows you to get to 16/17 instead of 19/20, and will lower your dmg/r across the board.
- All damage is given in expected values, done by hand. It's possible I failed my Intelligence(Statistics) check, so if you see something say something
- Humans used are Variant Humans, to take advantage of the bonus feat. Half-Orc is used otherwise as they're the only race I could find that can possibly add damage to their attacks through the Savagery feature, as well as giving a handy +2 Str.
- No magic weapons are used, as they are out of the budget of level 1 characters even if they sell all of their possessions, max out their starting gold rolls, and take the Noble background
- Using the point-buy system the half-orc takes a bigger hit than humans to both his to-hit values and his damage (the +2 doesn't get him to a higher Ability Modifier tier). This results in the Barbarian Polearm Master reigning supreme all the way to 24 AC, and the GWF picks up the slack at 25+, cutting the half-orc out of our equation completely.
- You want numbers? Here's a Google Sheet with some numbers.
Without further ado, here is my list of damage-optimized level 1 characters (please refer to the above sheet for numbers with AC factored in).
Human Great Weapon Fighter with Great Weapon Master feat (22.75 dmg/r, +1 to hit)
- +4 str bonus
- +10 feat bonus (-5 to hit)
- 2d6 (reroll 1,2) greatsword attack (22.333333 dmg/r)
- 5% crit (.41666666 dmg)
- Total damage on maximum Crit: 14+12+12 = 38 dmg
Human Barbarian with Polearm Master feat (20.3 dmg/r, +6 to hit)
- +4 str bonus
- +2 rag bonus
- 1d10 polearm attack (11.5 dmg)
- 1d4 bonus attack (8.5 dmg)
- 5% crit (0.4 dmg)
- Total damage on maximum Crit w/ both attacks: (6 + 10 + 10) + (6 + 4 + 4) = 38 dmg
Half-Orc Two Weapon Fighter (17.4 dmg/r, +7 to hit)
- +5 str bonus
- 1d6 shortsword attack (8.5 dmg)
- 1d6 shortsword attack (8.5 dmg)
- 5% crit including extra damage die from Savagery (0.7 dmg)
- Total damage on maximum Crit w/ both attacks: (5 + 6 + 6 + 6) + (5 + 6 + 6 + 6) = 46
Human Two Weapon Fighter with Dual Wielder feat (17.2 dmg/r, +6 to hit)
- +4 str bonus
- 1d8 longsword attack (8.5 dmg)
- 1d8 longsword attack (8.5 dmg)
- 5% crit (.45 dmg)
- Total damage on maximum Crit w/ both attacks: (4 + 8 + 8) + (4 + 8 + 8) = 40 Dmg
Pure Vengeance at 13th level
Stats:
Str 20 [17 base + 1 racial + 2 ASI @ 8th level]
Dex 10 base
Con 18 [16 base + 2 ASI @ 12th level]
Int 9 base
Wis 16 base
Cha 20 [18 base + 2 racial]
Feats:
Great Weapon Master @ 4th-level ASI
Fighting Style:
Great Weapon Fighting
There's not many choices to make with this path, if you're set with Vengeance Paladin. Pick up Great Weapon Master as early as possible and use it only on foes when you have a 85% chance to hit, (at 13th-level, that's 14 AC). Other than that, the -5 to attack hurts you too much as you'll do a lot more damage with smite than with the +10 damage of GWM. This is backed up here, see Markovchain's answer:
Larger base damage = Smaller benefit
This is not surprising. If you can dish out 30 damage without GWM/SS, and you use GWM/SS, then you are also risking dealing no damage on a miss by taking a penalty.
The higher your base damage, the more you have to lose on a miss. Whereas, the additional 10 damage is always constant, so it is more significant if your base damage is small.
In the case of you, being a paladin, and able to dish out 49 damage on a hit with Divine Smite, GWM is not that important to you but you should still pick it up for the potential Bonus Action attack on a crit or kill.
If you're satisfied with the above, here's a little bonus:
10 Eldritch Knight Fighter / 3 Devotion Paladin
A.k.a The Smite Knight
It's one of the highest burst damage I've ever seen played, and have had problems challenging as a DM. If you say everything is "flexible" and if you don't mind deviating from your original Oath and Class, this build is pretty solid.
It focuses on smiting each time you hit, and using Action Surge to double your first turn (It gets even more ridiculous at 20th-level, as you can then Action Surge twice). You can also keep on using GWM when you Channel Sacred Weapon.
The con about this build is that it requires too many ridiculous stats to pull off. But don't worry about that, you have the ridiculous stats needed to pull it off. Happy Smiting!
Best Answer
Since you have a number of criteria, I had to go through each one by importance to find the best build (or builds in this case). You can find a TL;DR below with a description of my methods after.
TL;DR
The best build optimizing:
Race: Envoy Warforged (secondarily: Simic Hybrid or Hill Dwarf)
Class: Ranger
Ability Scores: 8/16/16/8/16/8 (or 10/14/16/10/16/8 for 2 less EHP, non-Warforged also use this latter array)
Level 1: Magic Initiate (Druid): shillelagh spell
Level 2: Dueling Fighting Style, hunter's mark spell
Level 3: technically Gloom Stalker for uses of Wisdom, but Hunter and Monster Slayer are otherwise equal
Level 4: Polearm Master feat
Starting Equipment: (with just starting equipment, Hill Dwarf is a slightly better secondary race)
Scale Mail (not needed if Warforged)
Quarterstaff
Equipment Wishlist: (with this equipment, Simic Hybrid is a better secondary race)
Half Plate (not needed if Warforged)
Shield
First, the DPR without using major resources, here are the best builds for each class:
I allowed for a margin of 20%, so any build that got within 20% of the best build is eligible since it may have other criteria that make up for its slightly less resource-less DPR. I skipped Monk and Sorcerer because they offer strictly less than Fighter and Wizard respectively no matter what build you choose.
You can check out the class name sheets of this spreadsheet for some other builds which were elimintated as well as how these builds were decided
Bard
Cleric
The clerics include either magic weapon or elemental weapon
Druid
Fighter
Paladin
Ranger
Each build includes Hunter's Mark damage. I used Colossus Slayer as the Hunter's Prey choice for simplicity, other options are explored later
Warlock, Pact of the Tome
Each build includes Hex damage. I assumed that you could get one-fourth of your attacks on the target of Hexblade's Curse within short rests for the calculation. I assumed you could hit a second target with Green-Flame Blade on one-third of your attacks
Warlock, Pact of the Blade
Each build includes Hex damage. I assumed that you could get one-fourth of your attacks on the target of Hexblade's Curse within short rests for the calculation.
Wizard
I assumed you could hit a second target with Green-Flame Blade on one-third of your attacks
Now let's eliminate any class whose best builds can't step to the better classes
You can check out the "Classes" sheet of this spreadsheet for which builds were considered
Going down the list...
Our final builds (weeding out the weak Ranger and Warlock builds)
You can check out the "Warlock vs Ranger" sheet of this spreadsheet for which builds were considered, as well as which builds are better with which combinations of AC and advantage
After placing the Ranger and Warlock builds into the same pool, and only accepting builds that satisfy the 20% threshold of at least half of the Armor Classes and advantage-statuses between AC 14 and AC 20 (where most enemies will be at level 5), these builds remain:
These builds are in order of the number of AC/advantage-statuses they satisfy the threshold for, with more weight placed if they have the highest value and slightly less value placed on advantage
Second, the additional damage per day gained using other resources
Each of the five builds we arrived at have passable resource-less DPR, but they each have features that allow for other damage-increasing effects usually one or a few times before a short or long rest. It is all but impossible to evaluate which is best, so I will just give my thoughts:
Now, let's talk effective HP
Since all of these five builds are approximately equal when it comes to damage output, let's evaluate them based on your second criteria. The following table shows the effective HP (EHP) for an optimized version of each of our builds:
HW = Horizon Walker; GS = Gloom Stalker; MS = Monster Slayer
\begin{array}{c|r|r} \text{Build} & \text{EHP with best starting gear} & \text{EHP with best mundane gear} \\ \hline \text{Polearm Hunter} & 141 & 184 \\ \text{Polearm Hexblade} & 122 & 161 \\ \text{Elven HW} & 116 & 160 \\ \text{Wise HW} & 141 & 184 \\ \text{Polearm GS/MS} & 141 & 184 \\ \end{array}
I used this spreadsheet to calculate EHP, which assumes, among other things (source)...
As you can see, the Warlock build falls behind the Rangers when it comes to EHP. Furthermore, the Elven Horizon Walker can't keep up because of the race restrictions of Elven Accuracy.
If Warforged are not an option for your table, there are two races to substitute depending on what equipment you get:
Simic Hybrid (10/14/16/10/16/8) gets +1 AC from Carapace and results in this table:
\begin{array}{c|r|r} \text{Build} & \text{EHP with best starting gear} & \text{EHP with best mundane gear} \\ \hline \text{Polearm Hunter} & 123 & 182 \\ \text{Polearm Hexblade} & 84 & 159 \\ \text{Elven HW} & 116 & 160 \\ \text{Wise HW} & 123 & 182 \\ \text{Polearm GS/MS} & 123 & 182 \\ \end{array}
Hill Dwarf (10/14/16/10/16/8) gets +5 HP from Dwarven Toughness and results in this table:
\begin{array}{c|r|r} \text{Build} & \text{EHP with best starting gear} & \text{EHP with best mundane gear} \\ \hline \text{Polearm Hunter} & 125 & 174 \\ \text{Polearm Hexblade} & 89 & 154 \\ \text{Elven HW} & 116 & 160 \\ \text{Wise HW} & 125 & 174 \\ \text{Polearm GS/MS} & 125 & 174 \\ \end{array}
The armors for these non-Warforged builds are Scale Mail for starting equipment (Leather for the Warlock), and Half Plate for best mundane.
With the considerations of EHP, the optimal builds for your criteria are narrowed down to:
Although, I'd recommend against the Horizon Walker Ranger with the Dueling Fighting Style and +2 Wisdom since it is only as effective in damage output when Armor Class is 22 or higher.
Which Polearm Dueler to choose (Hunter, Gloom Stalker, or Monster Slayer)
We've addressed your two main criteria and come to the conclusion that a Warforged (or Simic Hybrid/Hill Dwarf) ranger with the Dueling Fighting Style and the Polearm Master feat is the most fitting. As we've shown, the three Ranger Archetypes Hunter, Gloom Stalker, and Monster Slayer all are relatively equal. First of all, you should choose the one that is more fitting of your campaign.
If you don't expect any of these to present itself more than any other, then we move to your bonus criterion:
If this is the last criterion, then Gloom Stalker is best.