I recommend traveling the loop clockwise. The cross-country section between Alta Mountain and Chikamin Pass is easier this way, the more challenging sections (in terms of navigation and terrain) will be in the first half of the loop, and the second half of the loop will be along relatively easy trails down Mineral Creek and along Little Kachess Lake.
The section between the tarns below Alta Mountain and Chikamin Pass merits a little more detail as well:
-traces of very old trail and the occasional cairn mark the way from below Alta Mountain until Chikamin Pass but overall it's effectively cross-country travel. The faint trail that climbs to Box Ridge is visible on the slopes east of the tarns below Alta and several of its switchbacks are hard to miss higher up. Once on the north side of Box Ridge, the trail starts out strong but then disappears quickly. I recommend continuing north towards Chikamin Pass, staying above treeline and traveling across talus fields, for the easiest route but it's also possible to drop to Park Lakes (through heavy brush) or stay high along the ridge (with much more scrambling and exposure).
Distance: 22 miles for just the loop but closer to 30 with the side trips, which are the highlights of the route
Elevation Gain: ~6,500' for just the loop and around 11,000' for the loop and both side trips
Maps and More:
A few trip reports describing the Alta Mountain-to-Chikamin Pass section: