Just FYI Skyline Drive tends to be crowded during that time of October. Little or no safe high-speed driving. The speed limit is 35MPH anyway and they do enforce. It is, however, a very scenic drive when the trees turn and certainly worth going on, just don't expect to be carving up any turns. If the weather is nice and the trees are in full color you will probably be lucky to even maintain the 35MPH speed limit for long.
By mid to late November it should be a lot emptier. During the middle of winter it's almost deserted, though there are still park rangers and deer, etc. so you still need to drive within reason.
Beware of going late to avoid the crowds. There is a LOT of wildlife that likes to jump out in front of your car. I've had several close calls at night even driving at or below the speed limit. If I'd been going fast I would have had a deer in the passenger seat.
Same holds true for the Blue Ridge Parkway. Personally I think the BRP is a better road than Skyline Drive, but Skyline is a much better day trip from the DC area due to location.
BTW it's $10/car to use Skyline Drive, so bring some cash.