Bear country, keep things clean and you should have no problems.
My favorite place is Big Meadow, almost like dispersed camping, no fee, not many campers, no water, yet fire rings and pit toilets.
It's between Kings Canyon NP and Sequoia NP on The Generals Highway (Hwy 198) then a northeast turn on Forest Service Road 14S11 (if you need more details than that just let me know). You do pass Big Meadow on the way to the campground units.
There are 5 campground units...
You'll see unit 1 first.
And, unit 5 is the most remote (roughly around these coordinates 36°43'21.82"N 118°48'47.84"W).
Lots of dirt roads to explore, and most of them can be done with a truck and FWC or ATC.
Easy to get to Wilsonia (private cabins, however, driving around is permitted, small store, too), Grant Grove, park headquarters, The Canyon, Lake Hume, (church camp, however, open to the public & gas available) and all of the main areas, yet usually peace & quiet back at your camp. If you leave your campsite, be sure to mark it "Occupied" (I leave a small tent), so no one will think you've left for good.
Enjoy.