You will probably want to do this route:
29 north and get on 28 east towards Manassas. Just before Manassas, get onto Route 234 Bypass north to 66, and head East to 28 again, take north to Dulles Airport. You will be going against traffic, and you shouldn't encounter any problems.
Some may suggest you do 29 through Warrenton and hook up with 66 in Gainesville, but I will advise against that because there is a HUGE road construction project going on to re-route the road over the train tracks, and there are still problems. A truck hit a train just a few weeks ago, and the entire area was a nightmare, and I think it was on a Saturday if I recall. Bottom line, avoid it, especially since you are trying to catch a flight.
Plan on 3 hours from Charlottesville so you don't feel rushed and will have wiggle room for any minor delays. You may encounter some slow traffic from the Chantilly area up 28 to Dulles, as you will be going with some rush-hour flow, but you shouldn't find it too bad.