Successfully complete the Batmobile level to unlock the time attack and extra missions under the bonus option.
Caped Crusader Suit
Complete the game on any difficulty.
Dark Knight Suit
Complete the game on any difficulty.
Defeating Ra's Al Goul
To defeat Ra's Al Goul at the end, make sure you defeat all of the ninjas that are around him; he will not die until all ninjas are dead. Also, when Ra's Al Goul is trying to attack you with the sword, it is best to use counter.