August is probably the busiest month of the year for Mt. Rainier- best weather before school starts and all. September is much quieter, but still excellent weather.
I've spent more time in the National Forest over there than in the Park itself...my favorite campspots are along the American River, to the east side of the mountain along hwy 410. There are a few dispersed camping opportunities along there, and several campgrounds. All accessible for a truckcamper.
Wildflowers are likely to be pretty much over except at high elevations....I did the hike at
this link once in late August and there were some beautiful small alpine flowers along the trail.