The short answer is

**Human Great Weapon Fighter with the Great Weapon Master feat wielding a Greatsword**

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, which makes me a bit relieved because rage is an approximately limited damage bonus.

You can also add Caltrops to this mix (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
 - I wasn't given an average AC, so I assumed all attacks hit. Including almost any amount of AC will **significantly** lower the actual dmg/r of the Human Great Weapon Fighter/Master
 - I gave all characters rolls of 18 for their stats, then added racial strength bonus. Point only allows you to get to 16/17 instead of 19/20, and will lower your dmg/r
 - All damage is given in expected values, done by hand. It's possible I failed my Intelligence(Statistics) check
 - 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.
 - 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

Without further ado, here is my list of damage-optimized level 1 characters.

Human Great Weapon Fighter with Great Weapon Master feat (22.75 dmg/r, +0 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)

Human Barbarian with Polearm Master feat (20.4 dmg/r, +5 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)

Half-Orc Two Weapon Fighter (17.7 dmg/r, +6 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)

Human Two Weapon Fighter with Dual Wielder feat (17.45 dmg/r, +5 to hit)

 - +4 str bonus
 - 1d8 longsword attack (8.5 dmg)
 - 1d8 longsword attack (8.5 dmg)
 - 5% crit (.45 dmg)