I work-camped at Yellowstone last season from May until the end of September at the visitor center at Old Faithful and have work-camped at two other National Parks in previous years.
It was a record breaking year for visitation with 3.7 million visitors already by the time I left. As everyone else has said the campgrounds fill up very fast (some of the ones run by the NPS are first-come-first-served but those are mostly dry-camping or elec only, the ones run by Xanterra all take reservations).
September is getting busier and busier for the national parks. More retirees are coming out then to avoid the families (and foreigners don't have the same schedules so they're visiting spring through fall) so that it's not as much of a way to avoid the crowds as it use to be. There will still be plenty of people, just a different age demographic.
Traditionally, Yellowstone's busiest months are: July, August, June, September, and May in that order. This year is the 100 year anniversary for the NPS and there are special events going on, so the NPS is predicting it'll be even busier than last year.
My best advice to avoid the crowds is to get up early and start your sightseeing in the morning. Any day of the week can be busy, but if you're on the road by 8 am you have a good three hours or so before the bulk of the day visitors drive in from Grand Teton, West Yellowstone, and Cody. Stay at Fishing Bridge so you don't have as far to drive to see the sights. Understand that Yellowstone is 2.2 million acres and that in good traffic it takes about two hours to drive from the north entrance to the south entrance. You're going to want more than one day to do it right.
Hope this helps.