TL;DR

POINT BUY
Fighter 13 DPR
Barbarian (already Raging) 13.65 DPR
Barbarian (had to Rage on first turn) 11.7 DPR

ROLLED STATS, vs 13 AC

Fighter 15.35 DPR
Barbarian (already Raging) 15.35 DPR
Barbarian (had to Rage on first turn) 13.26 DPR


vvv writefag starts here vvv

Level 1 Fighter V. Human
STR 16
DEX 14
CON 16
INT 8
WIS 10
CHA 8
Feat: Dual Wielder
(If rolling for stats, assign AS with the following priorities:
1. maximize STR mod
2. get DEX mod to at least +2
3. maximize CON mod
4. maximize WIS mod)

This build centers around having a horse at level 1, which is attainable if your DM allows you to use the alternate method of selecting your starting inventory, namely starting with gp and buying items rather than choosing items from a list.

Fighter starts with 5d4 x 10 gold, which averages to 125 gp. ask your DM if you can just take the average roll. I imagine they will say yes. if not, then this build is unfortunately susceptible to a bad gold roll here.

with your 125 gp from Fighter, purchase the following items:
50 gp - Draft Horse
20 gp - Two Lances
50 gp - Scale Mail
total price: 120gp

Thanks to the Dual Wielder Feat your AC is a whopping 17 while mounted and armed with your Lances.



Caveat
1. LADDERS. Or, more generally, actually moving the horse to where it needs to be. Your horse can squeeze into 5x5 hallways, but if you want to get it up a cliff, you're going to need to get creative.
2. Managing the horse. The rules are vague about when the DM should call for an Animal Handling check, and what the DC for those checks should be. It's also vague about what exactly happens when the mount gets 'spooked.' You'll want to work those out with your DM ahead of time. Further, there's the chance that your (rather expensive) horse will be killed in combat. Consider asking your DM whether your mount is allowed to make Death Saving Throws and expend Hit Dice during short rests. Remember to take the Animal Handling skill.
3. Possibly too strong. Your DM may simply decide he doesn't want to have a character with 13 DPR, 13 HP, 17 AC and 80 ft of movement fighting against his level 1 encounters.


Notes:
-Why not Barbarian?
At level 1 with a +3 STR mod, the Barbarian's Rage does add slightly more DPR than the Fighter's TWF Fighting Style (vs a 13 AC enemy, Barb gets 13.65 DPR during Rage, as opposed to TWF Fighter's 13 DPR.) Also, the Barbarian has resistance vs physical damage during Rage, which is clearly attractive. However there were a few reasons I decided to use the Fighter instead:
-Most combat lasts around 3 rounds. Barbarians enter Rage as a Bonus Action, which competes with the Bonus Action attack from the off-hand Lance. So in a 3-round combat, the Barbarian would actually only have 11.7 DPR due to missing his off-hand attack on his first turn. If your combat lasts at least 9 rounds, then the Barbarian finally catches up in DPR. Incidentally, if you rolled for stats and have a +4 STR mod, then the DPR from TWF catches up to Rage.
-Rage is a limited resource. Level 1 Barbarians only get 2 rages per short rest. If your DM runs more than 2 combat encounters between short rests, then you'll be weaker than the Fighter during some of those encounters.
-Money. The average roll Barbarian gold roll is only 50gp, which isn't enough to buy the Horse and two Lances. In order to actually buy all of the necessary equipment, you'll need to roll at least a 7 on 2d8, which is only a 20% chance. Also you won't be able to buy Scale Mail.

-Where to after level 1?
This build is optimized only for level 1 but I think it can be built into a powerful character with levels as well. Specifically a Barb/Fighter multiclass. At level 2, take a level in Barb to get resistance and +2 damage from Rage. Then take another point for Reckless Attack. Soon it will become trivially easy for enemies to kill your mount instead of targeting you, so consider either progressing to larger and more exotic mounts, or taking Mounted Combatant as quickly as possible so you can redirect all attacks away from your mount and make it Evasive. Probably take Barb to 5 for Extra Attack, then Fighter to 3 for Action surge and the subclass feature, then Barbarian until level cap. For level 20, barb17/fighter3 seems best since Brutal Critical has better synergy with your d12 and bonus attack, and the Fighter's Extra Attacks don't have great synergy with it and you honestly don't need that many ASIs for this build. Almost all of the subclasses for both Fighter and Barb can be helpful, but I would probably go Bear Totem for tanking (and talking to mounts!) and Champ for the Brutal Critical synergy. Whatever you do, don't go Eldritch Knight or Path of the Berserker. EK can't cast spells during Rage, and Berserker's Frenzy has no value while you're dual wielding.

-To make this build really broken, ask a friendly Moon Druid to serve as your mount. Or, have a friendly Bard or Druid cast Awaken on a wild animal. T-Rex cavalry, anyone?