Here's how I'd like to see it work:
Class - provides the PC's principle, adventuring-related abilities, ie to-hit and toughness for fighters, spells for mages and clerics, a set of highly specialized skills for thieves. These abilities are level-dependent.
Race - modifies the PC, both in terms of mechanics and fiction, ie elves get +1 to spot and usually like nature, dwarves get +2 vs. poison and usually like mines. These abilities are not level-dependent, but an option to gain more extraordinary racial abilities at higher level would be nice.
Background - establishes the PC's starting skills, in addition to grounding them in the game fiction, ie Black Bart used to sail on a pirate ship, so +1 to climb, rope use, balance and balance checks, plus, he might be recognized/wanted in certain ports. These are not dependent on level, either. Characters can gain more skills, but not an ever-escalating bonus to existing skills. Level-scaling abilities should belong to class.
Mix and match to taste.