Kings Peak is a classic Western U.S. state high point: over 13,000 feet high; not quite a climb but a rough scramble nevertheless; somewhat remote and set among beautiful backpacking country; and named after a now-obscure surveyor (Clarence King, in this case). Like other state high points, it is a popular ascent.
The Uinta Mountains claim to fame is it's the highest and most important (not the only) range in the 48 states running east-west. However, it should be more famous for offering perhaps the nicest wilderness country in the Rockies. Trailheads are reached at the end of long, dusty roads, and a huge wilderness area of layer-cake sedimentary mountains, flat alpine meadows, pristine lakes, and easy summits offers better than average solitude. Kings Peak is a good 12 miles from the nearest road, in the heart of the Uintas, and most parites will want to camp out, not just to avoid a greulling dayhike, but to relax and enjoy the backcountry a bit.
Kings Peak lies just south of the main east-west crest of the Uintas, the high point of a subsidiary ridge running southeast from the crest towards Mount Emmons. The west slopes are pretty steep and cliffy, but approaching from the east, or either the north or south ridges, is just scrambling.
Although located south of the Uinta divide, Kings Peak is usually approached from the north. From the Henry's Fork campground, accessed by a dusty gravel road from Lonetree, Wyoming, a trail leads south and up into Henry Fork Basin, a beautiful alpine bowl studded with small lakes and stands of trees. Kings Peak, looking like a shark's tooth, pokes its head above and between two unnamed summits in the basin's south wall. Most parties camp in the basin, then cross the divide at Gunsight Pass, descend a bit, traverse west, and climb Kings Peak by either the east face or the north ridge.
The peak can be approached from the south, too--its a longer hike, but offers the opportunity to clamber along the easy ridge between Mount Emmons and Kings Peak, featuring many 13,000 foot bumps. The Swift Creek Trail and Basin, north of Mountain Home, Utah, are probably the best approach for this route.