There is a pulldown menu in every weapon, including Unarmed that allows it to be associated with a skill. I have associated the skill, but it no matter how much damage I'm doing it doesn't actually seem to give me exp.
I know you can modify unarmed and that this works because I have a perk that multiplies damage if the weapon in hand has the Keyword WeapUnarmed, which I made up and added to the Unarmed item.

Another possible solution:
If anyone has some inkling of a formula that could be used to calculate how much damage is required to get a skill point, I can write a script dynamically attached to each actor that when they get hit if the weapon used is unarmed it modifies a global by the damage it took. Then if the damage passes over a certain limit, the player gets a ModFactionRank to a UnarmedStat faction I can make. Then I can write scripts and abilities to make the rest of the Unarmed perks.

Does anyone know how to associate unarmed with a skill?

i'm not sure about the actual "Unarmed" weapon, but i was able to make a different unarmed weapon (something like "brass knuckles") associated with One Handed and it worked fine, giving xp and skill increase

it was a while ago, so my memory is fuzzy, but if i remember correctly, the only difference between my unarmed and the vanilla one was that i gave it a damage value instead of 0 (which would make sense if the skill advance uses a damage based multiplier, you cant multiply anything by 0 and get an increasing value)

Nah, I've tried that. Raising the damage of Unarmed seems to do nothing. It bases unarmed damage off of your racial data.
Raising the racial data's unarmed damage doesn't seem to make experience happen either.
I've tried making a perk that would give a linear boost to the amount of EXP given by Unarmed attacks, no dice.
SetWeaponType to make it a sword does nothing either. In fact, I can't even get it to do anything at all.

I'm seriously running out of ideas. The only thing I can think of is that there is a hard-coded exception to normal combat rules for the Unarmed weapon. Maybe if I try duplicating it with a different name...
Couldn't hurt.

it works as a separate weapon guaranteed, so maybe you are right that there is a specific weapon override hard coded into the game for the Unarmed weapon

I think that is the case. I finally got this working by basically just editing the animations for a bound sword. If you name it unarmed, don't give it an inventory .NIF, make it undroppable, etc. it works like a charm. Now the problem is I don't think it scales damage with the UnarmedDamage AV. I can make a perk for that though.

EDIT: Wrong again, it totally scales with UnarmedDamage as long as the animation type is HandToHandMelee