Billy was Alice's fiancé.
After a big fight, Billy and Alice broke up. This moved Alice to tears, and Jervis Tetch seized the opportunity to win her heart. However, Billy returned and made up with Alice, which threw Tetch into a fit that snapped his mind.
Tetch used his mind control cards to make Billy break up with Alice again. He used Billy as Bill the Lizard in his Wonderland gang. He was the first to be released from the mind control by Batman. Since all the other members of the Wonderland gang were programmed to attack Batman, Billy could easily remove the hats of the others. Finally, he also removed the hairband that had enthralled Alice.