It has been done many times
The fastest such trip we know about included visits to all 50 states in about a week. But the two co-drivers never really slept...
If it is possible to drive through sections of all 50 states in three months -- but to really SEE what there is to see and do would take thirty+ years.
The best route I have ever seen for accomplishing the lower 48 was run by the Quest-4 Team. Their site and several other noteworthy attempts can be viewed by <a href = "http://www.roadtripamerica.com/links/travelogs.htm">looking at the travelog page<a/>. Some additional information about the focus of their quest is also <a href = "http://www.roadtripamerica.com/people/quest4.htm">available here<a/>.