I know in Oblivion you could pay certain NPCs to recharge a weapon for you, but I have not come across any in Skyrim who offer to do it. It seems like this feature was not added into the game because of the inclusion of the Enchantment skill. Paying someone
else to do it would not increase the skill. So you may as well buy filled gems if you want a quick way to recharge your weapon.
You recharge weapons with filled soul gems. You can buy them at the general merchant in Whiterun. Just view the item in your inventory and press RB to recharge it. If you have more then one soul gem in inventory it will ask you which one you want to use.
The stronger the gem the more it will fill the weapon.
What Darth said. Use a soul gem. Out of curiousity... what does Diablo 2 have to do with anything?? o_O I'd also like to add, that even using your soul gems to recharge weapons increases your enchanting skill, which I thought was pretty cool.