You'll only be able to play story mode for villians in the Distant Glory storylines. Distant Glory story mode is only unlocked after you finish Shade Impulse. Even so, you'll still have to train him to at least a level 40 to be sufficiently prepared to battle with the ghost characters that appear in the story mode.
There's also an option for playing Inward Chaos, which is a dangerous storyline mode that features opponents from a level range of 90 to 110. Unless you're skilled and/or decked out with good equipment, it's definitely not a place for levelling.
An alternative for levelling (other than quick battle modes) is to enter Duel Colliseum (also unlocked after defeating all of Shade Impulse). Fighting duel colliseum gives nice rewards and pits your character against other characters of varying difficulty to let you get used to their abilities and battle styles.