To unlock Poison Ivy, you need to go to the Hall of Justice and destroy the objects under the right side stairs/balcony (there's a Hall of Doom symbol there) to reveal a circular floor section that can be ground stomped with a large character. This will open the route to the Hall of Doom.
Outside the Hall sits Mr Mxyptlk who gives you the first quest in a series to defeat Lobo for a brick. Complete the quest then kill the quest giver to start the next in the chain of quests (if killing does not work like the Penguin, leave the area and come back).
The fifth quest is called "Love Stinks" handed out by Poison Ivy requiring you to use a magnet to create a potion. Give the potion to Ivy and she hands you the character token.
After this quest, the rest of the chain moves into the Hall.
The eighth quest is called "Five Studs Please" handed out by Harley requiring you to find three patients for her to treat. For completing this she gives you her character token.
Poison Ivy costs 25,000 studs and Harley costs 120,000 studs. Purchase both in the character selector and then switch to both of them at the same time (character A and B).