Entrance for best non-NPS campgrounds -
West Yellowstone is by far the best for FHU options, also for USFS dry camping and some limited utilities camping.
South has limited options for FHU camping.
East - no camping close and a huge climb to the park,
North East - beautiful amazing scenic highway - several hours drive to FHU campgrounds
North limited options for FHU
Time of year - I prefer September after Labor Day - families are largely gone and almost everything is open. Planning a early summer arrival is always iffy because of weather - my experience that whenever I plan a trip at the start of a season, that is the year the cold and snow lingers on into summer. The year I don't go early, snow is gone by May 1 and the place is crowded before Memorial Day.
One warning about September. Smoke from fires will always be thicker and obstruct move scenic distance views of some great mountains.
However, IMHO there is NEVER a bad time to go to Yellowstone. Yes, snow or smoke might restrict some views and access to some parts of the park, but whatever is open and viewable during any visit is absolutely amazing.