When you finish the Batmobile level, it automatically unlocks the time attack as well as the extra missions (they are under the bonus option).
Finish the Time Attack Modes with all other 21 characters.
There is a challenge mode, which has you obtaining gems and an obligatory time attack mode.
Other niceties include a Puzzle Mode, Time Attack Mode, Skin Edit Mode, and Mission Mode.
To get unlimited Nitros, hold the select button as you hit 'OK' at the track confirmation screen in single or time attack mode.