One week isn't really enough to see the whole High Sierra,...especially if you want to see the best of both Yosemite and Sequoia/King's Canyon. You'll be on the road all the time,...and, probably on the west side, where all the crowds are.
I would suggest selecting just one area and backpacking in for a week.
The area encompased by the High Sierra (typically considered to be that area that the John Muir Trail runs through,...about 200 miles in length) is really huge. You might read through this page: John Muir Wilderness