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

Background Theory

Broadly when choosing to optimise damage in a single round there are five variables you need to take account of:

• Average damage per round (I’m assuming that chance to hit has been rolled up into this variable)
• Damage variance
• Damage immunities/vulnerabilities
• Enemy HP
• Have you Crit’d (This fact significantly increases the math required.)

When you optimise DPR, you are optimising only one/two of these five potential variables (Criting can alter the ADPR).

If you are only rolling one dice, your damage variance on that attack will be relatively high.

To reduce the variance of the attack there are three broad strategies:

1. Increase the static damage for the attack (as in your firebolt vs dagger example) — Larger static variables weight it towards average damage.
2. Reduce the damage die (as in your firebolt vs dagger example) — Smaller dice have a lower variance.
3. Increase the number of dice you use in the attack (the more dice you roll, the more likely you are to get the average damage for a given attack) - This is an application of the law of large numbers.

In the situation where there is a damage type resistance/vulnerability then the choice between the two becomes clearer as you will be halving/doubling the result from the damage dice which will decrease/increase your DPR.

To answer whether or not you are more likely to beat a specific enemy depends on the HP, and variability in that enemies HP (determined by their hit dice). The same two considerations apply here.

If you are fighting an enemy with a large amount of hit dice, then by the same application of the law of large numbers you are more likely to be fighting an average specemin of the enemy.

Similarly if you are fighting an enemy with a large fixed HP component then you are similarly more likely fighting an average enemy.

Application to your example

The Firebolt has a range of damage values when it hits of 1-10 (with a 10% chance of each)

The dagger has a different range of damage values when it hits 4-7 (with a 25% chance of each).

A Goblin’s HP is calculated by rolling 2d6 (the 7 HP in the Monster Manual is simply the average value of this distribution). This gives the Goblin Monster a range of Hit Point values between 2 & 12 HP (obviously the extremes of 2 and 12 are pretty unlikely).

Both attacks have a 45% chance of a non-critical hit and a 5% chance of a critical hit.

The rest of this analysis will assume that we have hit (that makes the math slightly earier, and we can convert it into number of rounds by using this information later).

As a result, when we hit that translates to a 90% of hits are a non-crit, and 10% are a crit.

I’m also going to assume, for simplicities sake, that we are doubling the result of the dice when we crit, instead of doubling the number of dice we roll.

Vanilla Attacks

1 Hit to Kill

Taking your specific example of a Goblin (7 HP). When our attacks hit

• The Firebolt has a 40% chance of killing the Goblin in 1 non-critical hit (7–10)
• The Firebolt also has a 70% chance of killing the Goblin in 1 critical hit (4–10) x 2
• The dagger has a 25% chance of killing the Goblin in 1 non-critical hit (4 + 3)
• The dagger has a 75% chance of killing the Goblin in 1 critical hit ( 2 x (2–4) + 3)

Taking all of this into account, the 1 hit kill percentages are:

• Firebolt: 40% * 90% + 70% * 10% = 43%
• Dagger: 25% * 90% + 75% * 10% = 30%

3 Hits or more to Kill

At the other extreme, the chance of it taking more than two hits to fell the Goblin.

• Dagger: 0% (Even if we roll the minimum non-crit damage two hits will kill it.)
• Firebolt: We have to roll a total of 6 or less between the first two hits. With crits this is a more complicated piece of maths.

Ways we can get this total with Firebolt

2 x Crits:

• 1 x 2 + 1 x 2: ( 10% * 10% * 10% * 10% )
• 1 x 2 + 2 x 2: ( 10% * 10% * 10% * 10% )
• 2 x 2 + 1 x 2: ( 10% * 10% * 10% * 10% )

1 x Crit:

• 1 x 2 + 1: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 2: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 3: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 4: ( 10% * 10% * 10% * 90% )
• 1 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 2 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 3 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 4 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 2 x 2 + 1: ( 10% * 10% * 10% * 90% )
• 2 x 2 + 2: ( 10% * 10% * 10% * 90% )
• 1 + 2 x 2: ( 10% * 90% * 10% * 10% )
• 2 + 2 x 2: ( 10% * 90% * 10% * 10% )

0 x Crits:

• 1 + 1: ( 10% * 90% * 10% * 90% )
• 1 + 2: ( 10% * 90% * 10% * 90% )
• 1 + 3: ( 10% * 90% * 10% * 90% )
• 1 + 4: ( 10% * 90% * 10% * 90% )
• 1 + 5: ( 10% * 90% * 10% * 90% )
• 2 + 1: ( 10% * 90% * 10% * 90% )
• 2 + 2: ( 10% * 90% * 10% * 90% )
• 2 + 3: ( 10% * 90% * 10% * 90% )
• 2 + 4: ( 10% * 90% * 10% * 90% )
• 3 + 1: ( 10% * 90% * 10% * 90% )
• 3 + 2: ( 10% * 90% * 10% * 90% )
• 3 + 3: ( 10% * 90% * 10% * 90% )
• 4 + 1: ( 10% * 90% * 10% * 90% )
• 4 + 2: ( 10% * 90% * 10% * 90% )
• 5 + 1: ( 10% * 90% * 10% * 90% )

All of that is:

• 3 * ( 10% * 10% * 10% * 10% ) + 12 * ( 10% * 10% * 10% * 90% ) + 15 * ( 10% * 90% * 10% * 90% )

Which totals to 13.26%

Final Totals

As a result the probabilities of killing the Goblin with a repeated attack type are:

1 hit to kill:

• Firebolt: 43%
• Dagger: 30%

2 hits or less to kill:

• Firebolt: 86.74%
• Dagger: 100%

3+ hits or more to kill:

• Firebolt: 13.26%
• Dagger: 0%

Mix & Match

This is of course assuming you don’t mix and match attacks.

If we allow mixing and matching your choices change. For example, if you hit with the Firebolt or the Dagger for 6 damage, it doesn’t matter which attack you hit with next.

Similarly if you hit with the firebolt for 3 or more (80% probability for the hit), using the dagger next hit guarantees the kill.

If you hit with the firebolt for 1 on the first hit (10%) then your probabilities are:

• Firebolt: 50% (6 - 10) * 90% + 80% (6 - 20) * 10% = 53%
• Dagger: 50% (6 - 7) * 90% + 75% (7 - 11) * 10% = 52.5%

The Dagger’s smaller variance will probably give it enough to pip the Firebolt (but it’s close).

If you hit with the firebolt for 2 damage on the first hit (10%*90% + 10%*10% = 10%) then your probabilities are:

• Firebolt: 60% (5 - 10) * 90% + 80% (6 - 20) * 10% = 62%
• Dagger: 75% (5 - 7) * 90% + 100% (4 - 11) * 10% = 77.5%

You can see where this is trending…

Consolidated Percentages

1 Hit to Kill

• Firebolt: 43%
• Dagger: 30%

2 Hits or less to Kill

• Firebolt, then Dagger: (80%*90% + 90%*10%) * 100% + 10% * 52.5% + 10% * 77.5% = 94%
• 2 x Firebolt: 86.74%
• 2 x Dagger: 100%
• Dagger then Firebolt (we don't care as doing this would be stupid)

3 Hits or less to Kill

• Firebolt, Dagger, Dagger: 100%

3 Hits or more to Kill

• 3 x Firebolt: 13.26%

Strategy against a 7 HP Goblin

Hit with Firebolt first, as it gives you a better chance of one-shot killing the Goblin.

With two hits the probabilities of killing the Goblin are: Firebolt + Dagger = 94% Dagger + Dagger = 100%

If you hit with the Firebolt on your first hit, and it doesn't kill the Goblin (and the Goblin has 7HP) then you are better off switching your second attack to be the dagger, as you have a higher (and less variable) probability of killing the Goblin with the Dagger on the second hit.

Given that the increase of kill % with 1 hit on a Firebolt (13%) is greater than the decrease (6%) in % we get over two rounds from not doing Dagger, Dagger, Optimal play is to do Firebolt + Dagger.

On average how many rounds does it take for you to hit the Goblin?

We effectively have repeated trials to success of a iid Bernoulli random variables with p=0.5. The expected waiting time until a success (a hit) for this type of process is given by the expected value of a Geometric distribution, with p=0.5.

E[Rounds to Hit] = 1/0.5 = 2 rounds.

Thus the expected rounds to get 2 hits is 4.

Conclusion

This sort of spread will change depending on the actual HP of the Goblin.

• Lower than 7 HP will lean towards you using the Dagger for both attacks (6HP is the point where the one shot probabilities for both Firebolt and Dagger are close enough (53% vs 52.5%) that the consistency of the Dagger leans in its favour).
• Higher than 7HP Firebolt first (and potentially 2nd) Dagger later will be the better combo.