To unlock the Wayne Manor bonus level you must first complete all fifteen of the Super-Hero stud collection tasks in order to fully build the Hero Super-Kit that is found in the middle of the Hero Trophy Room.
Once it has been built you will be able to access the bonus level through a doorway between the Hero Episode 1 Minikits and the Hero Episode 2 Minikits. The Hero Trophy Room itself is accessed via a lift on the right side of the main Hero Mission Room of the Batcave.
Your task within the bonus level is to collect 1 million studs, there's absolutely no extra studs in this bonus level so you need to do every single thing in each of the areas otherwise you'll find yourself coming up short.
Head to the left along the front of the manor and you should find a set of large black barrels. Smash them and build a computer out of the bricks. Use the Technology Suit to switch the computer on and then get in the chair to start playing the side-scrolling game.
In this side-scrolling section you need to smash all of the cubes. You can smash the black ones by either by hitting them with a weapon or a punch, and you can smash the brown ones by hitting them with your head. Make sure to collect the purple studs left behind. Grey blocks are not smashable and some blue blocks are pushable or collapse under your feet or act as buttons to activate something. Then there's Silver blocks you need to explode and transparent blue blocks you need to shatter. In total you should have 470,000 studs by the time you leave via the pipe in the ground at the end.
After leaving the side-scrolling game you will appear on the roof of Wayne Manor where you can find a single 10,000 stud. If you get back onto the computer you seemingly just jumped out of you'll enter a whole new mini-game area, this time rolling exploding balls around blowing up fans.
To earn the 500,000 studs that are available in here you must blow up all of the fans, you can do that either by slowly nudging a bomb towards each fan and having it explode, or you can get in one of the green hover vehicles and drag a bomb around behind it smashing fans as you go. The vehicle method is quicker as the bomb won't explode for every single fan but can be trickier to get to do what you want. You'll probably end up doing a mix of both. Once you've reached 980,000 studs pull the lever in the middle to leave.
Leaving the ball game will drop you on the opposite side of Wayne Manor where there's another single 10,000 purple stud. That leaves 10,000 to go. To get that jump down and go to the right end of the area where you can build a green bulldozer. Use it to smash all of the rocks down on the grass and a gate will open. Go over to the opened gate and get on the lawnmower. Drive that over all the white flowers that have appeared on the lawn to get the final purple stud to appear in the place you just got the lawnmower from. Level complete.
Reward? A little bit of percent to add to your total.
In a similar manner to Wayne Manor to unlock the Arkham Asylum bonus level you must complete all fifteen Super-Villain stud collections to build the Villain Super-Kit in the middle of the Villain Trophy Room.
The Villain Trophy Room can be found by going to the right from the computer in Arkham Asylum and going through the door in the corner near the orange canister rather than the one next to it that leads to the Maximum Security Area. The entrance to the bonus level is up a ladder at the back of the Trophy Room.
Again you need to find 1,000,000 studs but this is laid out like more of a traditional bonus level with stuff to do just laid out around the level.
On the football pitch you can score three times each way to earn 10,000 studs each time and then a large silver trophy will pop out of the ground which you can blow up. Hit the punching bags a few times to earn a few purple studs. Run on the treadmill a couple of times to earn some purple studs. And smash all the rest of the gym equipment. I had 273,900 studs by the time I left this room but the level's forgiving so you don't need to have everything this time.
In the second area you're in a sort of maze with undefeatable enemies running at you. Go around smashing the objects and collecting the studs on the ground. Make sure to drive all of the vehicles as a trail of studs will appear when you get in each of them. Near the middle of it you'll be able to find a plant that you can take control of and start eating the enemies that are attacking you, they'll drop a few studs and respawn. There's nearly 900,000 studs in this area so you should be able to make the million without having to do everything, and if you're especially dedicated you can probably get the entire million just from this area if you keep defeating the respawning enemies.
Once again the reward is just some percentage added to your total. But by this point that may well be the last percentage points needed to bring you to 100% and unlocking that final character.