Ok, so good news and bad news about this.
Firstly, the good news - Mr Mxyptlk is pretty easy to get. From the hub, push backup and move the cursor on the globe right to the Halls of Justice. From here enter the halls and go right. Destroy the silver characters on the ground floor with Batman/Cyborg to reveal a broken floor - and use Killer Croc to smash the floor. This leads to the Hall of Doom. At the entrance, complete Mr Mxyptlk first two missions to unlock him. Don't forget to buy him in the character hub.
Now the bad news - Bat-mite is perhaps the hardest character to unlock. On every level you play he has a quest for you. In free play you need to complete his task (usually get something for him and bring it back). In addition, he also has a quest in the Hall of Justice. So over 15 separate missions, spread out over the entire game!!
In reality, if you are looking to 100% the game, then it's not a problem, as you will get it en route. However, if you just want this one, you have to do a hell of a lot of the game to get it.
Good luck!!