Boss mode will appear in the single player menu after completing either the Hero or Dark story, but you won't have access to all three boss attacks until you have completed all three stories.
Hero Side pits you against all the Hero story bosses, Dark Side against the Dark story bosses, and All, as you might expect, against every boss in the game, including The Biolizard and The Finalhazard. There are no breaks between these bouts so if you had any difficulty with any of the bosses you should stock up on lives before you start. Additional information regarding the bosses can be found here.
This will unlock after completing all three boss attacks.