Board Thread:Skyrim/@comment-11549959-20130717015134/@comment-11344160-20130717020604

Highlight the weapon you want to recharge in your inventory and an option should appear at the bottom of your screen to charge, prompting you to select a soul gem to use.

You fill soul gems by casting Soul Trap on an opponant before they die. This is a Conjuration spell (Apprentice level) which can be purchased at the College of Winterhold, or from the court wizard in Whiterun. The soul will automatically fill one of your soul gems as long as you have one large enough to contain the soul (examples: wolf is Petty-sized, average dragon is Common-sized, Mammoth is Grand-sized).

Soul Trap can also be found on Weapons, or Enchanted onto them. Creatures killed by these weapons will automatically fill a soul gem. A common practice is to give a follower a bow with Soul Trap enchanted on it and also give the follower a bunch of various empty soul gems.