This must be done in Free Play
There are two ways to unlock Joker, the first is to actually play the game's story mode, specifically the Villain story.
Joker's story is the third story sector in the Villain campaign and Joker will be unlocked for play by completing the first level.
Alternatively you can use a cheat code to unlock him, the use of the achievement will not effect your achievements so it is fine to use.
Joker (Tropical) - CCB199
The code is entered from the Batcave/Arkham Asylum computer hub.
Once Joker is unlocked for play head to the mission room and start up any Free Play level (That isnt a driving level)
Make sure you pick one of the Joker characters to be in your team.
Once the game begins press
until your character is a Joker, then press
to switch to the other player and also switch them to a Joker.
Now with both characters as Joker walk up and press
to use the hand buzzer.
Here's a video showing this.