Most of the stages will be visited as part of the core portion of the game, the Tournament of Legends. This 'story' mode has ten levels by default, and generally the same stages in the same order. Some of the levels might vary depending on your character; the achievement unlocked for me during the third playthrough with a third character.
Alternatively, twelve levels are available in Versus mode. You can play against the AI or other controllers, and you can pick a specific level you may still need in this mode.