Class skills require magicka
Class skills do not improve upon what weapon you wear and it's weapon dmg (tested)
Class skills' cost is reduced by wearing gear which reduces Spell cost
Class skills are therefor spells
Class skills improve with spell crit

If you want to improve the effectivity of your class skills, you wear full light armor
If you are only using Class skills and no weapon skills for some reason, you can run around with wearing no weapon what so ever. Surely you will lose the trait on this weapon that for example can affect your crit-chance, but like said the weapon dmg itself does not affect any Class skill what so ever.

This cleans up everything I had in my head regarding the typical approach to melee builds, like being a Nightblade for example and focusing heavy on the assination tree, it is pointless to wear medium armor. As much as it hurts to say this, as a fan of medium armor and the agility that usually comes with it. But melee crit does not apply to Class Skills. Spell Crit does apply to it. So aside from maybe the sneak detection or runspeed increase, it really makes no sense for someone focusing on Class Skills to run with anything other than Light Armor.