Think of Chin-Kee as Jin's/Danny's worst fears come alive. He represents everything they don't want to be associated with: the idea of the "chinky" Chinese. (Get it? Chin-kee/chinky? Clever, Gene Yang, very clever.) We're talking all kinds of sterotypes here, including things like: bad English (Chin-Kee can't get his r's and l's straight, as in "'Harro Amellica!'" (3.17)); yellow skin, slanted eyes, long queue, and buck teeth (3.1); and "model minority" behavior (like always getting the right answer in class (6.8.-6.19)).
Because Chin-Kee represents all the ridiculous stereotypes that haunt Chinese Americans, he also reflects back to the reading audience the racist nature of American popular culture. That's why Chapter 3 opens with a drawing that looks more like a TV sitcom title: "'Everyone Ruvs Chin-Kee'" (1). That title introduces Chin-Kee as a fantasy too—a fantasy produced by American pop culture.
Chin-Kee is useful though. Since Chin-Kee is so clearly Chinese while Danny is so clearly white, you've got to wonder how they could ever be real cousins. Is Chin-Kee a family friend? An adopted cousin? Nope—it's not only Chin-Kee who's fake (because he's actually Monkey); it's Danny, who's actually the very Chinese Jin.
But the only way we can get to this revelation of who Chin-Kee really is in relation to Danny is for the two to confront each other, which they do in an epic kung-fu battle (9.6-9.41). And the only way they get to that point is for Chin-Kee to be extra Chin-Kee, as in dance-on-a-library-table-singing-"She Bang"-Chin-Kee (9.5).
Chin-Kee outdoes himself so much that Danny's only reaction is to match Chin-Kee's outsize behavior with his own outsize rage. Hence: epic kung-fu battle scene.
So Chin-Kee's main purpose isn't just to make us cringe and confront our own racist stereotypes about Chinese people; he's really in the book to force Danny to become his true self, Chinese American Jin.