I agree with oomis suggestions 100%. Ride at a good pace and get alot of the mileage done in the first day. I did a similar trip in Oct 2011.

I suggest you start early and aim for Winchester, Virginia for the first day. It's about 20mins north of Front Royal and Skyline Drive. Also, in Winchester, there is a really affordable Best Western with Plenty of rooms, and they give you a voucher for the Perkins restaurant located in the lobby. (great b-fast!) Rooms were less than $70 USD.
Link to the BW:
http://maps.google.ca/maps?oe=utf-8&...&ved=0CKoBELYD

I stayed there twice! One enroute to BRP and once on the way back.

Leaving early in the AM, I topped up my 21 litre tank in Front Royal and hit Skyline drive in a ALOT of fog.

Riding hard and stopping only 3 times (1st to pee, 2nd to eat, 3rd for Fuel again) I just made it to Fancy Gap, VA before dusk.

The BRP is an incredible road and ALOT of fun but all warnings here should be considered.
-Dawn/Dusk rides brings out the animals.
-Fuel is NOT marked on the BRP so go on the website and print out the lists of Fuel stops by mile marker
-The BRP is difficult to do in a short amount of time.
-Don't skip any of the BRP. It's all amazing.

Good luck and have fun!