Even when traveling by car, I get around DC by metro and on foot. Parking is next to impossible or very expensive. Plus, the metro is a treat for the kids. I haven't camped in the DC area, so can't make any recommendations there.
As a native Washingtonian, I can tell you that's it really tough to stay close in. Very expensive at Cherry Hill...although it is the closest. I suggest looking in VA (Aquia has a nice CG) and taking the train/Metro into town. There is a KOA in NE Maryland, but it's kind of far out and not very close to buses and the Metro. As previously stated...parking is either non-existent or exhorbitantly expensive! You could pay MORE for parking for a day than you would for several days at a campground!!! Definitely try to go AFTER Labor Day!! All costs are lower at that point....although the Metro is quite pricey all the time!
Enjoy my hometown...it's a wild and beautiful place to grow up!!