This is where we go for the Summer (past 10 years). Upper Peppermint dispersed area in Sequoia National Forest. It's 14 day limit (free), but in all the times we've camped there, we've never been asked to move. During the week there might be 4 or five other campers by the river, but none by the road, the weekend can get kind of crazy down by the creek. We camp about 200 yards from the creek, and never hear any other campers. There are plenty of places to camp there where you'd never see or hear another camper. Camp fires are allowed, but you need to get a permit. I think the road is closed until Memorial day weekend. We've been up there as early as June 5th or so and there was still snow on the ground.
The limiting factor is potable water and black water tank capacity. There is no water on that mountain. We bring refreshments to the fire crew camp up there (about 2 miles from the cite) and they let us fill our 55gal water barrel. Otherwise, bring your own water up from one of the camp grounds in Kern. We use a blue tote to dump the black water and drive down to Kern when we need supplies. Don't forget to haul your trash down there too. In Kern there are grocery stores, touristy places, food, tube rental places, fishing, ect. In the summer it's hotter than hades (100 plus degrees) down the mountain. The river is cool, but it's packed on the weekends.
The National Park (where I grew up camping, Kings Canyon, ect) is actually really far away (I think my wife said it would be a 3-4 hour drive) from the National Forest. There's no easy way to get from one to the other. So this is not really a good jumping off point, unless you want to check out the Kern river.
The dirt road getting down into the dispersed area is a little rough the closer you get to the river. But you can park at the top of the hill (it's actually more level up there) and has better camping spots. The drawback is you'll be hiking down to the river. There is plenty for kids to do, miles of trails, swimming, biking, or just get lost in the woods. There's a little general store that has ice cream and a little diner about a mile from the camp. Bring your GOLD CARD though wink emoticon . On the mountain it stays in the 70's and 80's during the day. Thunderstorms are frequent though. There are no hosts or volunteers, no check in time, no trash service, no water, no sewer. There's a ranger that drives through maybe once a day. Sometimes you'll see the fire service guys run by. I've seen Fish and Game there, but other then that nothing.
It's super dusty there. The dirt is so fine it gets everywhere. It's like talcum powder fine. This last summer I sent the kids down to the river to get washed up from riding their bikes. When they got back they were as dirty as when they left. It's a little frustrating. So make sure you have a indoor out door carpet. Your dog and your wife will thank you.