Three public campgrounds we enjoyed on the Kenai were Hidden Lake CG at Kenai National Wildlife Refuge, Deep Creek State Recreation Area at Ninilchik, and Tenderfoot Forest Service CG at Summit Lake (not far from the turnoff for Hope). Of the 3, Deep Creek is the not-to-miss favorite and you can go charter halibut fishing right from the beach. Only problem with Hidden Lake was the young male grizzly that kept following people from the lake to their campsites. It was especially concerning after he raided the site behind ours because the idiots left a bunch of 12 packs of Coke inside a screened canopy over their picnic table. Needless to say, they needed a new canopy after that.
The only one you need reservations for is Deep Creek--very popular!