The story is that Ra's Al Ghul's daughter Talia tries to hire Catwoman to steal a crystal skull from museum but you refuse because she wants to steal it for herself. I have no idea why she wants it but apparently Catwoman is 5 years old and wants something the moment she finds out Talia wants it. I'll be honest, I really don't know much about the story beyond that because I didn't get far. The game isn't all that hard, but the controls are ass and nothing about it really made me want to keep playing.
It's kind of a shame, I mean on one level kudos to DC for licencing out a second stringer character for a game, and not just Batman or Superman (although this is sort of a Batman-ish game) but putting this on the Game Boy Color was a mistake. If you would have put this on the Playstation you might have had something. You could even have kept it 2D, in fact most of the 2D PS1 games like Mega Man 8 or Castlevania: Symphony of the Night have aged better than the clunky polygons of their 3D contemporaries.
This game isn't awful, and it's certainly not a bad as it might have been, and I guess if you're a hardcore fan of the character it might be worth checking out, but for me the fact that they made it was far more interesting than the game itself.