The term Chinaman is also slang for henchman. Generally someone who can do dirty work for you. Most likely the orgins of this is that 19th century Chinese laborers were known for doing harsh and dangerous jobs (e.g. building the railroads).

This meaning of the term is the whole joke in The Big Lebowski. While mentioning the Chinaman that peed on his rug, The Dude was talking about the guy's job. Walter thought he was discussing his nationality.

I believe the proper term is Asian-American, Dude.