Champion fighter would definitely be the dip, 3 levels max. That way I'd get the improved critical and the action surge. I'm thinking maybe Halfling Barbarian Berzerker with a 3 level Champion Fighter dip. It would take until level 6 to start to work. At 6 I'd have 3 levels of each which would give me reckless attack, action surge, frenzy, and improved critical. So once in a frenzy I'd be able to make two attacks a turn with advantage and the ability to re-roll ones.
At level 8 I'd get the second attack as a 5th level barbarian which would translate into 3 advantaged attacks in a frenzy, 5 with an action surge. I like pairing the easily obtained advantage of reckless attacks with the expanded crit range and basically eliminating ones (which increases the crit chance).
I know a lot of people pair champion fighters with hexblade warlocks but I'm wanting the character to just be a crazy combatant with essentially no magic.
I've also considered a champion fighter/rogue build using an elf or half-elf. Champion fighter dip to 3, elven accuracy feat giving the extra d20 on dex based attacks with advantage, then rogue getting advantage attacking from being hidden. So that would be 3 dice to crit 19-20 plus sneak attack which I believe doubles as well.