We have used Pohick Bay twice and it has worked very nice for us. Wooded and not expensive. Good base camp for visiting the DC area. Just go to the first Metro stop and take the Metro into town.
Click on this for a link to a webpage some generated that tells you about all of the DC area campgrounds.
I live about 15 minutes from Greenbelt Park and we used to use it on weekends that we couldn't take a road trip. It is a popular park in the summer but in early April you should have no problem.
It's a clean, safe campground that is close to downtown, whether you drive it or take the Metro. The Metro is probaly your best bet. It is user friendly and the Orange line takes you right to the center of the National Mall outside of the Smithsonian museums.
Dave
Mary, the world's best wife (1951-2009)
Isabel & Buddy (the Bostons)
2005 Itasca Suncruiser 35A
DC was a fly in for us in 2008, so can't help on the campgrounds (though I will keep the Greenbelt in mind when my retirement trips come around)
We stayed on the outskirts of DC, at a hotel with a Metro station beneath it. Very helpful. DC isn't a place I'd suggest driving around, especially if you're trying to visit the Mall. The Metro will drop you off closer than you could hope to park. We ordered a 7 day Metro pass for everyone before we got there, didn't rent a car until we were ready to drive south.