In the lower reaches, you will find larger browns coming up from the Yellowstone. Streamer fishing to these browns in the fall is your best shot at a fish of some size in the Clark's Fork. Runoff will put the Clark's Fork out for June and most of July. Once that is over, in the upper section you will immediately begin to see caddis and this will continue through the summer. Both Green Drakes and Gray Drakes appear in late July and run into August.
Hoppers can be good in the hot weather.
Standard nymph patterns work through the season. That said, a handful of Royal Wulffs and you're set for the summer. Along with the Wulff, you will see locals relying on Stimulators and Humpies. The canyon section, which is best known for its white water with category IV rapids, is generally fished with large attractors and nymphs. The fish here will get only a quick look at the menu so you might as well make it highly visible.