We backpacked Browns creek and Little Browns Creek Trail in the beginning of August. There were great spots for camping along the way- some of the best camp sites I've stayed at. If you start the loop going 'clockwise', there is a wonderful first campsite with a view of a lake and the mountains. There is water (streams, waterfalls, lakes) near and on the way to this campsite. I suggest filling up here, as water is sparce/hard to find beyond this campsite. Our last campsite just past Mt. Anterro was my favorite. With a view of the valley at one side and mountains at the other… it was breathtaking. Highly recommend! There was some evidence of bears, so I'd suggest planning accordingly with bear spray and bear bags. There is also no camping allowed at the trailhead.
We stayed in the backcountry windmill area… at site 8 I believe. It was one of the farthest back sites and was a great one! This is the type of place to call and ask which site is the best because not all of them were equal. There is a great loop you can do around the entire park if you want to backpack it all. It was very hot when we went, we recommend going on a cooler day ;)