I've driven it a couple of times in each direction... I've thought about riding it since I am on the east oast currently...
From memory, it is 105 miles each way...nothing is flat. It can have traffic or it can be vacant. As all National Parks go, you'll have to pay an entrance fee and there will be tons of RVs and the like.
I recommend the blue ridge parkway...no fee, less cars and the same terrain....less scenic overlooks and facilities, but I think it would be a more pleasant ride. I'm also a big fan of West Virginia for hiking and the like, even fewer people than Virginia and the terrain is similar...
Good luck and enjoy...
T