Here's my very-favorite trip that I used to do every summer. It's the Sierra High to the John Muir trail, then back down.
You start in Onion Valley along the Sierra High Route (outa Independence), head up to Kearsarge Pass, then meet the John Muir to Rae Lakes. It's fairly short, but beautiful, so I enjoy giving myself at least 1 day without any hiking involved when I'm in King's Canyon. You can get over Kearsarge pass in one day pretty easily, but I love camping at Flower Lake and almost-always break it up into two days, then camp the second night at Kearsarge Lakes (and spend an extra night there). Then from Kearsarge Lakes to Rae Lakes (which is kinda brutal).
It's a steep climb that goes above 11,000 feet, with a heck-of-a-lot of up and down. Parts of it are above the treeline, and if you wanna go offtrail, there are some really pretty things to see. You will get shit from rangers if you get off-trail around Bullfrog Lake. The view from Kearsarge pass is amazing, and camping at the lakes offers some great cliff diving spots.
Here's someone's youtube footage around Rae Lakes:
https://www.youtube.com/watch?v=clrA1IwvVGY 