Remember: America is big and many people live in this country our whole lives and don't see everything worth seeing.
Remember, things are far apart. Just for fun, I plugged in a route for such a trip (starting in Boston and ending in Seattle, with a trip through the South as planned) and came out with about 70 hours of drive time. If you can manage about 10 hours a day driving, this means that 1 of your 5 weeks will probably be taken up by drive time, so just be prepared for that.
For the record, I chose:
Boston
New York
Philadelphia (I might actually skip this leg and come down through New Jersey. It get's a bad rap, but once you're in the south, there are some nice beaches. Then again, the shore is going to be crowded in the summer, and if you like history, there's some great stuff in Philadelphia).
Baltimore/DC (these two cities are next to each other. DC has a lot of our national monuments, but Baltimore has some great food).
Nashville (it made no difference to do Atlanta instead, time-wise, but I prefer Nashville for the music scene. If you go, don't go to be big venues, that's going to be mostly pop-country. You want to get to the rough part of town and find someone playing the blues. Also, you get to go through the mountains to get there, so that's cool)
New Orleans (This is one of the great American Cities, and despite the way they talk about it, it's still standing and it's still fun. Great for music and some of the best food in the country)
Austen (If you're willing to take the long way, you can drive along the Gulf to Galveston, and then up through Huston instead of making a beeline for Huston. Huston, by the way, has some cool stuff but isn't as nice as Austen for my money. Either way, Texas is dull. The part before Austen isn't quite as bad from what I can tell, but after Austen, you will likely pas through stretches of long country roads and only country music on the radio, so come prepared with your own music).
Flagstaff, AZ (Near the grand canyon. Before you get there, you'll pass through the Gila and Apache national forests, which also are quite nice).
LA
San Fransisco (California, like Texas, is huge but there are slightly more interesting things to see most of the way. After SF, you get into Northern California, which has some great national forests you may wish to check out).
Portland
SeattleI've never been to either of these last 2, but I hear they're fun.