I prefer Limestone and send most inquiries about camping there first. The bad news is it's NOT on the National Reservation system and you can't make reservations, you won't know if any spots are available until you get there. The good news is it's NOT on the National Reservation system. Try there fist and if nothing available work your way down river. Fairview is next best and has a general store with and excellent restaurant. If you want closer to town try the private Rivernook Campground.
Although they're threatening closing dispersed (free/little or no facilities) camping spots due to public degrading the areas, there is still dispersed camping along the river above Riverkern. Make sure you camp at least 25 feet from river bank in established open areas only. Report littering, vandalism, cutting vegetation and unsanitary practices to Tulare Sheriff 559-733-6211 or US Forest Svc. 760-376-3781 Good luck and stay safe.
For upper Kern campground into visit:
http://www.kernvalley.com/news/kerncamp.htm