Any time from a week from now after the water drops through the end of July.
Grab a Forest Service map to determine what is private and what is National Forest on river right. River left (actually, everything East of the center of the river) is Glacier National Park, and you need a backcountry permit. Most private boaters avoid this hassle and just camp right.
Good 4-day trip: Border to Ford Cabin, Ford to Polebridge, Polebridge to Big Creek (NF campground), and Big Creek to Glacier Rim (whitewater stretch). Like catflipper said, you could stretch the upper part and get in an extra day, or you could run longer days and do it in 3 days. Big Creek to Glacier Rim is only ~2 hours of river time depending on flows, the other stretches are 12+ miles with a bit less velocity.
It is bear country, so plan to hang your food and camp smart. Bring bear spray.
I'm possibly interested, I've never done it as a full trip with overnight camping, we have always done day trips and stayed in FS campgrounds (or at home!) when our kids were younger...now I want to get them out for some fun overnights.