I just got my first class artifact for my Siren. On the skills/powers page, I notice it says it is level 1 in the description and the number 1 appears beside its icon. Does that simply denote the level of the item, like owning a level X required weapon, or is it something that can improve somehow?
You will find higher level artifacts (up to a level of, I believe, 6) through gameplay as rare drops from enemies/chests. You will only keep the highest level drop of a given element (explosive, fire, corrosive, shock) per character. I'm not sure if there's any trick to finding artifacts as drops, they seem to occur like any other uncommon drop.
None of the items you pick up can level up. Once you get them, they stay at that level. This also applies to Borderlands 2.