How much time do you have? There is so much to see on that route that I'm sure you'll have to pick and choose.
We've made that trip a few times. Here are some of our favorites.
Cape Kennedy, St. Augustine, Savannah's Topke Island, Myrtle Beach, Cape Hatteras and many more like that.
If you are into history there is a campground at Harper's Ferry, WV. that has civil war entrenchments within the grounds. It's a KOA and to my way of thinking a good place to base your drive into DC.
Also Appomattox and Sharpsburg, Gettysburg, Bull Run, ect.
The Smithsonian can easily take a week if you want to see it all.
There is also a large amusement park between Richmond and DC that has camping. I believe it's called Kings Crossing.
Of course there is Williamsburg, Jamestown, and Yorktown.
You may need to make more than one trip.