I posted this 20 minutes ago, and the BPL server seems to have slept through it. Let me see if I can remember my text.
Many backpackers enter Kings Canyon from the East Side, and the Bishop Pass trail is one common entry point. Some climbers choose a point a few miles west and northwest of there, Echo Col. Starting from the East Side, typically [edit: Lake Sabrina], you get up to Moonlight Lake and spend the night. Then get up early and pass Echo Lake while the permanent snowfield is still firm. From below Echo Lake, you can't really identify where Echo Col is, but I have it marked on a map. You may need to get off the snowfield to the right side and scramble a bit.
One you get through the notch to the West Side, it gets a bit more challenging, Class 3, I believe. A single backpacker could make it, but it is easier with two people so that you can hand down your backpacks. I would not do this with a heavy backpack. You can go directly down to the John Muir Trail or you can contour to hit the JMT closer to Muir Pass. It depends on which direction you are headed.
I went through there west to east about 25 years ago, and then east to west about 12 years ago.
Edit: I photographed Echo Col from the west side just a few days ago:
As you stare at this rocky crest, you still wonder where Echo Col is, exactly. In the photo, I believe it is on the right, about 20% in from the right edge. The rubble below the col is smoother since people have walked over it. As I recall, it was much easier to find from the east side.