Kings Canyon itself is a bit of a cul de sac but is adjacent to Sequoia NP But given the time you will be in the vicinity, I would recommend exploring the groves in the Sequoia NP area. For a days drive Mineral King is worth a look see. The road in is 25 miles with a reputed 700 turns but very easy grades and no vertigo involved. Campsites are available at the end of the road but without services.
The whole area has great hiking and backpacking.