Danny sees Chin-Kee as the living embodiment of all the many offensive stereotypes that have adhered to Chinese Americans over the years. His whole identity, from his yellow skin and his buck teeth, to his chronic inability to pronounce his r's properly, appears to have been constructed by the most virulent white racist imaginable. As Danny doesn't want to be associated with such a walking, taking stereotype, he keeps his distance from Chin-Kee.
One further aspect of Chin-Kee's stereotypical behavior is his exemplary behavior at school. Much to Danny's annoyance, Chin-Kee is a smart kid, who always seems to get the answers right in class. A lot of children don't have much time for students who do well at school; it makes them feel inadequate. And Danny's no different, especially as Chin-Kee's wizardry in class perpetuates an additional stereotype of Chinese Americans as being a model majority.
But Danny can't distance himself from Chin-Kee for long. At some point, he's going to have to confront him in an epic kung-fu battle that will finally reveal who Chin-Kee really is. (Spoiler alert: he's Monkey, the heroic figure of ancient Chinese legend.) In doing so, he will also be forced to confront his own true identity as a Chinese American called Jin.