Is there a way to level up unarmed combat (punching)? I found a perk but I want to get better and level up!
In Skyrim, there is no class skill to level up unarmed combat (or Hand-to-Hand). Simply punching things won't help you.
There are, however, ways to improve your unarmed combat.
Kahjiit/Argonian: You Start off with a passive 20 damage if you are Argonian, and 25 damage if Kahjiit to unarmed attacks, these races make the unarmed warrior the most powerful threat early in the game.
Heavy Armor: I take the full tree since you are going to be right up in the enemy's face at all times, but you only need 30 points to get the Fist of Steel perk that will make any gauntlet's armor rating your attack damage.
Smithing: Armor is your weapon. Improve your armor for more damage.
Enchantment: This is what will make you truly deadly. Once you get the power to enchant you will be able to add Unarmed bonuses to all your armor and become a wrecking machine.
Go to riften and start the thieves guild stuff by working with the guy named brynyolf and doing his quests plant the ring and go to the ratway, in a room with an alchemy table and a lot of floor traps a guy runs up and starts punching you, no matter your race his gloves add 10 damage to your hand to hand.
If you're an Argonian you start off with 10 Passive damage and Kajit 15 Passive damage. So The first thing I did is went to the Companions in Whiterun and leveled up heavy armor with Farakas.
Next I got the "Fists of Steel" perk and then you wear your strongest heavy armor gauntlets and the armor rating of the glove/gauntlets is your damage added to the Argonian or Kajit damage.
Alternatively, look at applying one or more of the unarmed mods.