Early October might just beat the peak time for leaf viewing of Fall foliage, but expect lots of traffic on both roads, especially on weekends. Local cyclists know what to expect and find alternate places to ride. Expect bumper to bumper traffic getting in and long skeins of cars and motorcyclists, RVs and tour buses throughout the park, be especially careful around overlooks where people can pull off to view vistas. Ride early and get off the road by noon, when city people have finally gotten to the parks and beaten the back ups at the entrance stations. These are not quiet country roads with only local traffic. Wear bright colors, use both head and tail lights, wear your helmet, keep as far to the right as "practicable."
While the grades are not particularly steep they are long. The North entrance to Dickey Ridge is a five mile climb with 4-6% grades and more of the same along the full 108 miles of Skyline. The descents can be wonderful, of course, and make the climbs worthwhile. BUT we locals know that both wildlife and traffic are significant hazards on fast descents. Deer are very common, unhunted, tame and are unpredictable. If you see a deer (or bear) by the road do not assume it will not step into your path. If you want to enjoy wildlife along the road then ride early in the morning. As I said, deer are common, but bear and wild turkey can be frequent early. Far less traffic, too.
I'm not sure when we turn the clocks back an hour so keep in mind the amount of daylight you have to get from place to place. Lights are required by law after dusk and low light conditions will prevail during inclement weather. Use lights that illuminate the road ahead as well as make you more visible. Fog is common on these roads. It will be chilly at night and in the mornings and if it rains. I have never encountered a nice warming rain on Skyline; it always gets cold. Going fast downhill when soaked to the skin has its own perils. Dress for heat and cold.
On the other hand, with the exception of some forever to remain unnamed roads, Skyline and the Blue Ridge are two of America's best bike tours, deserving of their reputations as beautiful and challenging. Savor every mile.