"via the Blue Ridge Parkway". If you get on Skyline Drive just south of Winchester VA and follow it to the Blue Ridge Parkway and then follow the BRP all the way to Asheville, you will either be in a state of Zen-like calm or ready to kill someone after what will seem like endless days of driving 35-40 mph around one curve then another.
I would either a) drive to Asheville via I-95 to I-81 to I-26, get here in about 11 hours, THEN pick out stretches of the BRP to drive en route to and from Mt. Mitchell, Craggy Gardens, Mt. Pisgah, Graveyard Fields, Sliding Rock, Du Pont State Forest ; or b) pick out a multi-hour stretch to drive on the way down and a different stretch on the way back. Everybody has their favorite stretches between Grayson Highlands in Virginia and Cherokee in North Carolina...I like them all. Depends on what you like, e.g. Blowing Rock is great for (expensive) touristy shops and restaurants, Little Switzerland is great for a enjoying a cold beer or glass of wine while the sun sets over mountains 50 miles away, Graveyard Fields and nearby Black Balsam are great hiking ,etc.