There are restrictions on where you can camp in some of the Wilderness areas. Part of the Great Gulf is off limits to camping - but you shouldn't be there. The same goes for the Tuckerman's ravine area. Again, probably not part of the itinerary. No camping in the Alpine Zone. You will be there!
The real problem is going to be finding trees large enough to hang from - there is no much there along the high ridges. Even down in the spruce forests it can take a long time to find trees available to hang on. (Tent camping areas are harder to find - but there are a few. )
I am familiar with some of the areas you may be hiking through. If you narrow down your trip I might be able to help you find areas to camp in.
I believe that most people either stay in the huts, or do the high ridges in a day when traversing.
And do watch the weather - it can get rough up there quickly. Know where the escape routes off the ridge are and don't be too macho to use them. Sometimes just getting a few feet below the ridge can really get you a big change in weather.
Here is a list
of the camping regulations. I can help interpreting them if you are not familiar with the areas.
One possibility is to hike up from Randolph on day 1 - stop while there are trees and camping areas. On day two do the high ridges. If you continue down the main Presidential Ridge you will run out of trail before the end of 3 days. The Montalbaum ridge would give you room for more hiking, or you could descend into the Dry River valley.