16 of 19 update to-hit numbers, sheet was updated at this point too

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 Pole Arm 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
  • 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