There are some places to boondock, more the further north/northwest you get. There are also plenty of state and provinical/territorial/municipal campgrounds that typically have a fire ring and picnic table for $10-$20 per night, dropped in an iron ranger. Many do not have potable water or a dump station, though pit or vault toilets are generally available and quite often non-potable water. The Yukon provincial campgrounds offer free campfirewood as well.
In the larger cities and bigger tourist destinations, it's somewhat harder to find really cheap or free camping, but there are options available in most places that are not too outlandishly expensive.
Church's book ("Traveler's Guide to Alaskan Camping") is an excellent and thorough guide to camping accomodations.
I would definitely suggest taking the trip, and reserving as much time as practical for it! Depending on exactly where you end up going, it may be rough on your trailer, but IMHO totally worth the wear and tear. It seems to me a bit of a truism that the very best camping experiences, at least in that part of the world, are on some of the least smooth roads.