This depends a bit on what time of year you plan to do this trip and whether you value time more than scenery. The most direct route from New Jersey
to Estes Park
goes out along the Pennsylvania Turnpike to ClevelandChicago
This would be a 27 hour trip and you could reasonably do it in 3 days or less depending on how long a driving day you want. That would mean you needed to pay for only two nights on the trip.
It has been a really long time since I drove this route and I have no camping options to offer. If you drove as far as Toledo on the first day, there is a Days Inn near the airport with a price of $47/night. When you get farther west near Omaha, you might be able to stay at a Rest Area off the Interstate. Or you can look on Google maps along the route and see what campgrounds there are.
But this route, while direct, is relatively flat once you get past Pennsylvania (except for around the Mississippi) And Kansas and Oklahoma are kind of boring to drive through. It would be possible (if you are not traveling during the winter) to drive down the Blue Ridge Parkway in Virginia and there are campgrounds there. But doing that route is going to take longer.