Late winter along the BRP
To the best of my knowledge, the BRP never officially closes in winter. At any point in time, however, snow and ice will close particular segment(s) since snow removal efforts, equipment, and budgets are all in short supply. The Asheville-Blowing Rock segment stays at fairly high elevation--4,000' to over 5,000', so it is in fact more frequently closed than other, lower, segments to the north. The National Park Service (NPS) maintains a recorded message, updated frequently, concerning road closures. Closures are also reported in real time on the NPS official website for the BRP.
Recognizing a strong bias for my native state, I'd most certainly recommend accessing the BRP in Asheville. Doing so gives you the Asheville-Blowing Rock-Doughton Park segment, ending right at the NC-VA state line, and that includes some of the highest elevations and most dramatic topography anywhere on the Parkway. You may also take in the short spur drive to Mount Mitchell, the highest point East of the Mississippi. It's a drive-up leading to a 100 yard paved pathway accessing a large concrete viewing platform at the summit. If you have good weather on that first day, you should by all means spend the hour, hour and a half summiting Mount Mitchell.
Irrespective of the real possibility that April may bring reasonably warm weather to the High Country, your views will be entirely of gray, bare trees. It's good due to the leaves otherwise blocking some views, and many of us happen to find the more monochromatic vistas interesting in their own regard.
Taking into account the likelihood of a Mount Mitchell side-trip and the relatively greater number of spectacular pull-offs to enjoy, I'd consider a slow and very leisurely 120 mile first day, from just southwest of Asheville at MP 393 to Blowing Rock at MP 295 (or thereabout, I forget the exact milepost at US 321). Overnighting in Blowing Rock has the advantages of being within a mile or two of the BRP, choosing from some nice and downright quaint motels, and some very good dining and refreshments all within a few block's walk along Main Street. Check out the live webcams at www.resortcams.com for a quick peek at downtown Blowing Rock. If you get in to town early and the weather is nice, spending some time on the Town Park benches overlooking Main Street can be very relaxing.
A 6 hour second day, averaging 35 mph including stops, would bring you to Peaks of Otter, VA. There is a Parkway Visitor Center, lodging, and a large restaurant at Peaks of Otter. A fee-required short bus trip can be engaged to take you to the summit of Sharp Top, and that's a worthwhile investment of time and money.
From Peaks of Otter, it's right about 200 miles up the remainder of the BRP + Skyline Drive (SD) to Front Royal, VA, the northern terminus of the SD. Again, with somewhat more widely distributed turn-outs, this can be driven in about 5-6 hours, thus enabling an early day stop at Front Royal or a longer reach up to Frederick, MD.
The NPS has a printable BRP map on the official website. Some advanced study of it and having it on board as you travel would be indispensible in my opinion. The map indicates mileposts (MPs) at key road crossings and attractions and also depicts nearby state and Federal highways nearby. These may be needed as bailout points in the event fog or ice and snow set in.
Enjoy your Blue Ridge Road Trip!
Foy