To orient you some: west coast of US tends to have the lowest humidity, so even hot weather is more bearable than hot and humid. The Southwest is a more desert climate, higher elevations, lower humidity, but hot!
The Pacific Northwest is absolutely beautiful that time of the year, and traveling towards Yosemite, Yelowstone Natl Parks is a great family tour. Lots of beautiful vistas, leg stretches and open land. The upper Midwest (Minnesota, Wisconsin, Illinois, Michigan, Iowa) are nice places, but they can get very humid and hot at times. Not as much to see, but all beautiful places with some certain things worth the trip.
The Central US in the summer is a bit sketchy. Hot, and often high likelihood of tornados. I'd stray further south or north to avoid these areas between June-Sept in the U.S.
Down thru Texas, hot, humid, and wet. But well worth a trip to traverse the west-to-east course. And you'll find open travel, good travelers accommodations, and plenty of members down that way to share stories and beers with you. But an eye on the weather patterns is a must. Barber Museum and Vintage bike week is a great idea. It's in Alabma, very hot and humid, so plan accordingly. Easy to reach from Texas.
If you did got to Barber, I'd turn north east from there and follow the Smoky Mountains for some great riding. I'd avoid the Tail of the Dragon as often it's filled with chucklehead wannabe sport riders wreaking all sorts of havoc on travelers. It's a technical ride, and maybe not worth the effort. Plenty of other just-as-great riding in the Smoky/Blue Ridge Parkway. I think Prokop just rode this route last year to and from Washington DC to Barber.
DC is wicked humid, but if you're keen on US history, it's a must. Else, treat it like a bad case of syphilis. NY is further north (about 600km from DC) and a great trip, but traveling on a motorcycle is not how I'd recommend it. Stay further west, and skip DC, NYC and go to the Poconos, and points north (Vermont and NEw Hampshire White/Green Mountains). Less humidity, great rides, an beautiful vistas. Older, earlier style of settlements in the U.S., plenty of historic stuff, and summer is the best time of the year for seafood in New England (healthy 2-4# Maine lobsters) and lots of fresh clams.
Finish in Maine on the sea coast (Atlantic ocean). Some cool, old light houses, traditional fishing village culture, and a completly different dialect of English than you've ever heard anywhere on earth, except for maybe Wales.
Traffic along the west coast edge of California is indescribable. You'll never reach your destination, and should be avoided at all costs. Use interior routes, thru different states if you choose to do a North/south west coast thing. It's beautiful vistas, especially Northern California, but traffic is full stop 24x7 and will cause you enormous stress, especially in a camper van. Or on a bike.
If you wanted to get from California to the East coast, you have 3 routes that are straight shots: the infamous Route 66 (farthest north) Route 40 (nearly dead central, to slightly south of central) and Route 10. 10 cuts across Texas and into Florida at the Atlantic Ocean. Turn left and use 95 to go from FL to Maine. (I wouldnt, but it gives you a perimeter of the states). Roughly 5,000km from West Coast to East Coast.
So there's a little bit of detail to help you rule some things in, or out and acclimation on where things are.
