Episode Bonuses are unlocked after completing all six levels on an Episode in Story, and having a certain amount of Gold Bricks.
Every Bonus door has three challenges in them: Super Story, Character Bonus and Mini-Kit Bonus.
The Super Story involves completing every stage in an Episode again. To get the Gold Brick for it, you need to beat both the best time and the best score. This can be annoying, as if you beat one but not the other, you need to beat both again to get the Gold Brick. Upon a New Game, the scores are set at 1 hour, and 100,000 studs. This means you have about 17,000 studs and 10 minutes per stage. The easiest way to get ready for this is to practise stages in Story that you are unfamiliar with, and practise both being fast, and collecting a decent amount of studs at the same time. These can be difficult, especially if you only beat one record but not both. You also cannot use Extras in this mode.
Character Bonuses and Mini-Kit Bonuses involve you being situated in a small area, based on the Episode you're playing, and you are tasked with getting 1,000,000 studs. These aren't that difficult as the amount of studs dropped by objects is increased tenfold, enemies drop studs, and we also have the introduction of the elusive Purple Stud, worth 10,000. In Character Bonuses, you play as normal characters, while in Mini-Kit Bonuses, you can play as the Mini-Kits you've completed. Extras can be used here, except for the Stud Multipliers. In general, Mini-Kit Bonuses are easier than Character Bonuses.