First of all Mineral King is a hard place to get up to. The approach road is about 25 miles long, but mostly single lane with plenty of sections with a fatal fall on the downhill side and a rock face on the uphill side. So it'll take at least an hour, a nerve wracking, puckering hour of driving to arrive. Once you're there most of the cars are wrapped in tarps from tire to roof to protect them from marmots. With the bad news out of the way we can get to the good stuff.
Head up the Timber Gap trailhead and follow all the intersections towards Sawtooth Pass. At about 5 miles in you'll get to a pit toilet with one of the finest sunrise views available, try to time this right and enjoy the show. The next mile is going to be rough, it's a sandy slog to the top of Sawtooth Pass and the last mile has about 1,200 feet of elevation gain, good luck. Once you're at the crest you'll head down between Columbine Lake and its little unnamed, uphill neighbor and then drop into a real gem of the High Sierra, Lost Canyon.
This is probably the most runnable section of the day, just about 4 miles of California carpet downhill trail, it's gorgeous. You're going to start working uphill once you cross Lost Canyon Creek towards Big Five Lakes Basin, DO NOT MISS THIS LEFT. Beyond BFLB is Little Five Lakes Basin at the top of this basin on the Great Western Divide is your second major pass of the day, Black Rock Pass. Enjoy the views of the Kaweah Crest, consume some calories and catch your breath because you have a long, rocky descent from here. The first couple miles are steep but very runnable. The lower you get the rockier and more technical it becomes though, don't expect to make up a lot of time on this section, it isn't as fast as it looks.
At the bottom is Cliff Creek, cross this for the final ascent up to Timber Gap and then you have just over two miles back to the trailhead.