The basic idea is a normal Barbarian rages and gets stronger. This Barbarian gets so mad, he throws down his weapon and punches you.
My main idea was using TWF with your fists and Dragon Style to make punches far harder than a weapon, but with variant multiclassing, the normally low feat requirement for a Barbarian suddenly jumps very high. It will be 9th level before being able to get Power Attack, Dragon Style, and Raging Vitality, with Dragon Ferocity coming in at 15th if I don't pick up TWF.
I know that most of the damage from attacks comes from static bonuses, but it doesn't feel like a raging barbarian using his fists if he is just rolling 1d3s.
I think you would be better off going something like Monk (Unchained) or Brawler and then doing VMC Barbarian.
From a thematic perspective, if you want to make his thing punching then there is no reason for him to have a weapon in the first place. A Barbarian that uses a weapon isnt going to get so mad that he throws his weapon down and starts punching, he would just use his weapon to kill people.
Personally i would go with a 1 lvl dip in Unarmed Fighter: it gives you Improved Unarmed Strike and Dragon Style as bonus feat.
If you do want to stick with Barbarian doing UAF 1 / Barbarian 19 is probably the way to go, with no VMC (honestly the monk VMC is pretty awful).
Also if you are worried about feat space, doing Unchained Barbarian takes away the need for Raging Vitality, and doesn't change much else, except for 2h weapon users losing a couple points of damage per attack.
Well you know upping the 1d3 is just your enjoyment...
Unless you enlarge there aren't a whole lot of ways to get damage dice up higher. A monk's robe will get it up to a d8. Ascetic Strike will get it up a bit, but they don't stack. An enlarge spell, or Stonefist Gloves which are kind of the same thing.
Really the problems are going to come down to a couple of things. You are going to pay twice as much to enchant your weapons and have to invest more feats to maybe break even on damage. Probably not even that when you consider you could be using a Falchion and have a 15-20 crit range. But if you are set on doing an unarmed barbarian build, go for it.
What Genoin said.
I think UW has a barbarian archetype with IUS.
Although I would just spend two levels on brawler. No alignment restriction, no need for double slice, and it counts as having two weapon fighting for the purpose of future feats.
If you don't like martial flexibility, then get mutagenic mauler (or warrior, I can never remember the name) to nab a mutagen for +4 strength that stacks with rage.
TWF is better the more bonus damage per hit you have, so something like rage + mutagen + dragon style + TWF should make for very fun full attacks.
Strapped for feats? Replace barbarian with the viking fighter archetype. Now you get full barbarism rage AND bonus feats, with the option to take rage powers instead.
