Have you checked the MetroRail system map? Attached.
While I haven't looked for apartments in a while (been here since '78), you may be lucky enough to be arriving at a down time for real estate/rentals, and able to find some particularly good stuff.
Capitol Hill might be 'fun,' but safety-wise, I wouldn't stick my neck out to suggest it. No Metro in Georgetown. DuPont Circle area very busy, socially; good luck finding a place there! Adams Morgan busy too; check it for safety. U Street not as metro-rail accessible as it should be. Foggy Bottom/George Washington University might be interesting.
Have Fun!
http://www.stationmasters.com/System_Map/system_map.html:hi: