On the West side of the park there is Timber Creek CG. It's first come-first served. Weekends are pretty full, but during the week there are usually sites available until mid-afternoon. Plenty of trails in the area, and access to the Colrado River.
As for time of year, I like it all. Now you would see the baby Elk, Moose, and Deer. In September you'd see and hear the Rut. Summer the nights wouldn't be quite as cool.