The most well-known Chinatown by far in New York City (there are several) embraces areas that were once inhabited by Italian, Irish, Eastern European, and other immigrant groups. Historical residues are everywhere. You can get there on multiple subway lines; I usually take the N, Q, or R to Canal Street.
This extensive community is bounded very generally—according to Google in early 2013—by Broadway on the west, Essex Street on the east, Henry Street on the south, and Hester, Canal or Grand Street on the north, depending on how far east or west you are. (You can search on “Chinatown Manhattan NYC” and look at the Google map yourself to get a better sense of the neighborhood’s range.)
Today, especially around Canal Street, Chinatown is quite the tourist trap, but unlike its Little Italy neighbor, this remains a major residential neighborhood as well.
If you wish to avoid people energetically hawking their wares, don’t spend too much time on Canal Street itself. I tend to end up more south and east, and the further south and east I go, the further I seem to get from the touristy stuff.