As previously stated 20002 and 20003 are Capitol Hill zip codes, but the neighborhoods change within those zip codes. Capitol Hill is primarily the 5 or 6 city blocks closest to the Capitol building. Capitol Hill East starts around Eastern Market at 7th Street and extends all the way to the Anacostia River. Within that is the Eastern Market neighborhood, Barney Circle which is around the Congressional Cemetery (named for where the old trolley car system used to have a turnaround/circle), Klingman Park/DC Armory which are north of East Capitol Street and extends up to the H Street neighborhood/Atlas District. South of Eastern Market is the Navy Yards area which extends to the I395 border.
If you want to see the breakdown of all zip codes in DC, or anywhere, this is a great website to use: http://www.hipcodes.com
As Patrick said, the two zip codes on the Hill are 20002 and 20003. The southern half of the neighborhood is 20003, and the northern half is 20002. However, the neighborhood changes pretty significantly from west to east, and so those changes can't really be mapped out using zip codes. As a Hill resident, I'm super familiar with the area and would love to show you my neighborhood! My contact info is below - please feel free to reach out to me directly if I can be of any help.