Use Khajiit and plan on using heavy armor. The heavy armor skill tree has a perk that gives you an unarmed bonus based on the armor rating of your gauntlets, or if you choose light armor, there are magical gloves somewhere in Skyrim that also benefit your unarmed skill. Believe me, it's completely feasible to play the whole game just using your fists (maybe not so much against dragons and other bosses). Even as a level 30 Khajiit, I deal more damage with my naked hands than with the best sword.
I'm not sure about this, but does unarmed damage ignore armor ratings? It seems like it does.
Look into how unarmed registers its damage.
The above posters all have the right idea. I would DEFINETELY go khajiit. Just for that unarmed bonus (sorry argonians)
Just check what damage it deals and adjust based on what info you get.