TL;DR: 21.805 DPR vs an AC 13 creature
Variant Human Hexblade Warlock with the Inheritor background and Polearm Master feat. We assume 18 in Strength because other answers did so, although V.Human (+2) and point buy (15) can't quite reach this - so you'd have to roll to get these stats (which is RAW)
You sell most of your starting gear (except for the armor and component pouch) to purchase a Scroll of Find Familiar, a Poisoner's Kit, and a Glaive.
Summon a rat. Collect as much poison as possible. In combat have the rat use his action to apply poison to your weapon. Glaive + Hex + Poison + Polearm Master attack = 21.805 DPR against 13 AC.
The build:
First, a baseline for how the existing answers fare against 13 AC - the average at level 1:
- V.Human GWM Fighter: 10.05 + 0.4 (crits) (assumes you use GWM for -5 to hit)
- V.Human PAM Barbarian: 14.21 + 0.4 (crits) (assumes you're raging, a twice-per-day mechanic)
- Half-Orc TWF Fighter: 13.05 + 0.7 (crits)
- War Cleric GWM: 18.9 + 0.7 (crits) (only benefits from War Cleric for 4 attacks per day, then DPR is cut in half)
Next, we'll have to make some assumptions about how a normal adventuring day will play out. You'll see why later. For now, we'll assume the following:
- You will engage in 2 battles per short rest
- Each battle will last 5 rounds of combat
- In combat it will take 2 rounds to kill an enemy
- You will rely on stealth to initiate every fight
The last bullet point allows us to assume Hex is cast at the start of the fight. We don't use it to factor in surprise rounds or advantage.
Here's how the build starts:
- Start Variant Human (+2 CHA, Polearm Master feat)
- Roll stats (We're assuming 18 STR, like the other answers)
- Hexblade Warlock (Hex)
Warlock grants us the benefit of Hex. This lasts an hour and can be transferred between enemies, so it will generally last one or two combats. Furthermore, your spell slots reset on a short rest, so we can assume nearly 100% uptime in most campaigns.
Now we need to choose our background and starting equipment.
As a Warlock, your starting gear will be:
- (a) a light crossbow and 20 bolts (sells for 13g) or (b) any simple weapon
- (a) a component pouch (keep) or (b) an arcane focus
- (a) a scholar's pack (sells for 20g) or (b) a dungeoneer's pack
- Leather armor (keep), any simple weapon (choose light crossbow and sell for 12.5g), and two daggers (optionally keep)
That's 45.5 gold from your class. For your background you want to choose Inheritor, who starts with 70 gold. That's a total of 115.5 gold to buy our starting gear. Buy the following:
- A Glaive (10gp)
- A Scroll of Find Familiar (50gp, see below)
- A Poisoner's Kit (50gp)
- A backpack (2gp)
There are various rules on the cost of a spell scroll. Per the DMG a magical item of common rarity costs 50-100 gold, and consumables cost half as much. This puts the cost between 25gp and 50gp. We'll summon a Rat as our familiar. He'll live in our backpack. Familiars move on your turn and can use any action a normal player can use, including Use an Object.
A Poisoner's Kit allows you to extract poison from unconscious or dead creatures for free, so long as you spend 1d6 minutes doing so and make a DC 20 Nature (Intelligence) check. This means that given enough time to prepare for an adventure, you could theoretically have infinite poison. Just locate a Giant Centipede, Giant Poisonous Snake, Hellwasp Grub, Awakened Giant Wasp, or similar. These creatures all have poison that deals 3d6 damage on a failed DC 11 Constitution save, or half as much on a success. They aren't even that hard to find, as the Monster Manual explicitly states a Giant Poisonous Snake can be found in the Desert, Forest, Grassland, Swamp, Underdark, or Urban environments.
Combat plays out like so:
- At the start of combat you will apply poison to your glaive and Hex an enemy
- Every round thereafter your familiar with come out of your backpack to re-apply poison to your weapon
- When an enemy dies, you will use your bonus action to re-target your Hex onto a new enemy
- If the enemy you're attacking is not dead, you will use your bonus action to make a second attack with the butt of your glaive
Damage Calculation:
With the assumptions we made previously, we can now see how combat will play out:
- Action: 1d10 + 1d6 + 3d6 + 4 = 23.5, Bonus: 1d4 + 1d6 + 4 = 10
- Action: 1d10 + 1d6 + 3d6 + 4 = 23.5, Bonus: Retargeting Hex
- Action: 1d10 + 1d6 + 3d6 + 4 = 23.5, Bonus: 1d4 + 1d6 + 4 = 10
- Action: 1d10 + 1d6 + 3d6 + 4 = 23.5, Bonus: Retargeting Hex
- Action: 1d10 + 1d6 + 3d6 + 4 = 23.5, Bonus: 1d4 + 1d6 + 4 = 10
That's a total of 147.5 damage over 5 rounds, or 29.5 DPR before factoring in accuracy. With 70% chance-to-hit (assumes 18 STR) that's 20.65 DPR before crits.
To calculate crit damage we take only the damage from dice and multiply by our crit chance. 115.5 dice damage, divided by 5 rounds, times 0.05 = 1.155 crit damage.
20.65 + 1.155 = 21.805 DPR (against an AC 13 creature)
Recommended Progression
Now that you have an optimized level 1, what about the other levels? Turns out, these are super strong, too!
- Level 1 is clearly described above, although one variant you might consider is buying a Quarterstaff instead of a Glaive. Since it's one-handed it can benefit from Hexblade to use CHA instead of STR, eliminating the dependence on two stats. You will switch back to Glaive at level 3
- Level 2: Warlock again, pick up Agonizing Blast (improve your ranged damage fallback) and Repelling Blast (good crowd control, and allows you to attack while "disengaging")
- Level 3: Warlock again, Pact of the Blade. Swap Agonizing Blast for Improved Pact Weapon. This increases your DPR as you gain a +1 to-hit and a +1 to damage. A summoned Glaive can replace your level 1 weapon
- Level 4: Warlock again, pick up War Caster to help maintain concentration on Hex and gain Eldritch Blast as an opportunity attack, which comes into play next level
- Level 5: Warlock again, pick up Thirsting Blade for the Extra Attack feature. We'll also swap Repelling Blast for Agonizing Blast as Eldritch Blast has now become the better option for opportunity attacks (making two beam attacks instead of one Glaive attack)
- Level 6: It's finally worth it to start multiclassing. Go Paladin, as Smite provides good nova to complement our poison, and the class will provide some of the best long-term benefits
- Level 8 (Paladin 3): Pick up Oathbreaker, giving access to Control Undead (another body hitting things = more DPR)
- Level 12 (Paladin 7): Aura of Hate. This turns your undead helper into a slightly stronger undead helper (plus it buffs your own attacks)
- Level 14 (Paladin 9): We finally get Animate Dead. Surrounding yourself in 8 skeletons. All skeletons benefit from Aura of Hate. You're now a whirlwind of death
- Level 15: Back to Warlock, giving us Spectre (which also benefits from Aura of Hate) and letting us swap out (now useless) Thirsting Blade for a more useful invocation. I recommend picking Repelling Blast back up.
- Level 16: Warlock again, by this point we've likely found a +2 Glaive, so our Improved Pact Weapon may also be worthless. Swap it out, and pick up two invocations (one to replace Improved Pact Weapon, one because we've reached Warlock 7). I recommend Relentless Hex and Eldritch Smite.
- Levels 18 (Paladin 11): Two more levels of Paladin. It takes us until level 18 to reach "tier 4" instead of 17, but by level 18 we receive Improved Divine Smite - granting+1d8 to all attacks! This helps us keep pace with Fighters who will soon be doing 5 attacks per round to our 3
By level 18 your DPR should be something like:
- Your main attack: 1d10 (Glaive) + 1d6 (Hex) + 3d6 (Poison) + 3 (Magic Weapon) + 5 (CHA) + 5 (Hate) + 1d8 (Smite) @ 70% to-hit = 25.9
- Your second attack: Same, minus the poison (7.35) = 18.55
- Your bonus attack: 1d4 (Butt) + 1d6 (Hex) + 3 (Magic Weapon) + 5 (CHA) + 5 (Hate) + 1d8 (Smite) @ 70% to-hit = 16.45
- Your skeleton army: 1d6 + 2 + 5 (Hate) @ 20% to-hit x 8 boney bois = 16.8
34.2 of this is from modifies and not dice (mostly Aura of Hate), so crits add 2.14 DPR
That means at level 18 you're seeing 77.7+2.14 DPR. This beats out a level 20 baseline Fighter (57.8125 DPR). So this build not only had the best level 1 damage, but remains good the whole game!
You also have 2 levels to spare at the end (Warlock 7 / Paladin 11 / X 2), so feel free to put a cherry on your build with something like Peace Cleric, Clockwork Soul Sorc, etc. You could even pick up Druid if you wanted to add some fun utility with Wild Shape (like finally becoming your own snake for the sake of gathering poison)
Final Note
Truthfully, this is all a little bit optimistic since it assumes the poison will do the full 3d6 every round. On a DC 11 the poison only deals half damage, so approximately 3.675+0.2625 of the calculated DPR has a chance to not land.
This means the level 1 DPR is actually between 17.8675 and 21.805 (depending on the creature's Constitution save modifier)
The level 18 DPR is less affected since most of the damage comes from Aura of Hate rather than poison. Its modified DPR is between 75.9025 and 79.84 (depending on the creature's Constitution save modifier)
Despite 17.8675 (our worst case DPR, against a creature with a +11 Constitution modifier) being slightly less than the War Cleric's 19.6 DPR, I still believe this is the superior build because:
- Our damage is consistent throughout the whole adventuring day (not just 4 attacks and we're done)
- Unlike the Cleric, we have a good ranged option (Eldritch Blast)
- After level 5 we have a stronger Opportunity Attack option (War Cast + Eldritch Blast)
- I never factored in Hexblade's Curse, which grants us a huge bonus to damage against one enemy per day
- Thanks to Aura of Hate, our damage scales better into late game