Kings Canyon opens usually third Friday in April. If you arrive that day you will find maybe 10 to 20 other campers in the entire area. It's my favorite time of the year to go there, most of the trails will be open for long distances and no annoying crowds (not that Kings Canyon is ever that crowded).
There are over 400 campsites in Cedar Grove but they only open as needed. In late April they only open the Sentinal campground. Last time I was there I took up one of the 80 campsites and five others were occupied. You'll have a lot to yourself!
If it's open, you should take the Boyden Cavern tour. It's at the west end of the canyon. Website:
http://www.caverntours.com/BoydenRt.htmWhen you're done in the canyon I recommend the short self-guided tour through Grant Grove and then take the excellent drive to the General Sherman tree in Sequoia Park and, if conditions are right, climb Moro Rock if you're not afraid of heights.