The easiest way to get to Yorktown/Williamsburg from Dulles is to take a domestic flight on to Richmond (50 miles from Williamsburg) or Newport News (15 miles). There is a shuttle service from Richmond on to Williamsburg, or you can cycle from there.http://www.groometransportation.com/
Bicycling from Dulles is not impossible but would be a challenge. Should you decide to try, go west out of the terminal, look for Old Ox Road (RT 606) to get you started in the most uncongested direction. Avoid Rt 28! You might find a campground near Manassas. (Bull Run Regional Park?) Google Map it on to town of Independent Hill and you will be on the Adventure Cycling Map #34 of the Atlantic Coast Route. 100 miles south brings you to Ashland and the junction with the Trans America bike route. (There's a rather expensive KOA campground on that leg back just south of Fredericksburg). Go east from Ashland on Trans Am maps #147-150 87 miles to Williamsburg. Yes, the same route you'll be coming back west on.
Bear in mind that the roads even beyond outer suburbs of DC can be a nightmare if you take the wrong one. Once you get on the AC routes they meander but they will take the quietest possible route.
The Tidewater Potomac route east from Fredericksburg is nice and more direct and would save you riding the TransAm route twice to and from Ashland. But it will dead end you at a long narrow bridge over the Rappahannock At White Stone. You might be able to beg a ride across from a motorist, then some busy roads and more bridges from there on the final leg to Williamsburg.
Good luck! Starting the Trans Am is the hardest part!