To unlock new cups (as well as obtain Chaos Emerald cards) for Cup Circuit mode, you have to complete Easy, Medium, and Hard challenges in specific zones in Challenge Mode as follows:

Effect - Code

Blue Chaos Cup - Complete all challenges for Death Yard Zone
Cyan Chaos Cup - Complete all challenges for Colosseum Highway Zone
Green Chaos Cup - Complete all challenges for Forest Falls Zone
Grey Chaos Cup - Complete every challenge in the game
Purple Chaos Cup - Complete all challenges for Meteor Base Zone
Red Chaos Cup - Complete all challenges for Sky Park Zone
Yellow Chaos Cup - Complete all challenges for Crystal Mountain Zone