Are you in a position where you can drive around the area a bit before signing a lease? Or do you pretty much need to know where you're going to live before you get here?
Try Oxford Square Apartments.
LERNER RESIDENTIAL - OXFORD SQUARE
You'll see from the availability listing that it is not supercheap, but it's not bad for the area and (unless I have my apartment complexes mixed up), you'll be in a fantastic school district. You'll be assigned to Garret Park Elementary in Kensington, MD, Tilden MS, and Walter Johnson HS. All really good schools. And while there will be some kids there from affluent families, you won't be overrun by them like you would be in some other clusters.
I'd also suggest going to the Washington Post's classifieds and looking for people renting out their own places. Once you have an address of a place, you can look up what school it would be assigned to by plugging the address in here:
Division of Long-range Planning
That won't, of course, tell you anything about how good the schools are, but generally, any school in the HS clusters of Bethesda-Chevy Chase, Wootton, Richard Montgomery, Winston Churchill, and Walter Johnson are schools I would be comfortable sending my 7 year old to.
When in doubt, go to
GreatSchools.net and search for the school you're interested in. There will often be comments posted on that site from parents about their experiences there. My kid transferred to a different school after we butted heads with the principal at his old school (and spent about $10,000 in lawyers' fees), and greatschools.net has a bunch of negative comments about the principal. They're accurate, at least in the context of my son's old school.
His new school? Phenomenal and a great principal.