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Sounds like a long time for that lake. Camping is allowed only at designated sites, even in the backcountry.
There are lots of regs to comply with. Check with NPS on the entrance pass regulations, as the normal park pass is valid for only one week. You will also need to get a boating permit (sticker) for a small additional fee.
Backcountry sites must be reserved, either well in advance by mail, or within a day or two of the start of your trip in person at the backcountry office. You are not allowed to just occupy a site--you MUST have and display the permit for that site. To obtain the permit, you must first go through their backcountry orientation, which is done in person at one of the backcountry offices. When you have completed the orientation (which lasts less than half an hour), they give you the actual permit.
If this sounds like a lot of paperwork, it is (but not compared with trying to get river permits elsewhere). In return for your efforts, you know that you will have campsites, and nobody else is allowed to squat there. The sites usually have a bear crossbar and pit toilet. Generally, you will not be able to camp on Yellowstone Lake till mid- to late July due to bear considerations.
Bring bug juice and/or bugsuits.
I'm not going to tell you which sites are best, as that is extremely subjective, and the choices depend on how many miles per day and other schedule variables.
There is one guidebook to paddling in Yellowstone and Grand Teton, a good one. Pnet sells it, or at least used to.
Might have better chances of avoiding people during the off-season. At least then you won't have the big outfitter groups that book the best sites way ahead of time.
The catch is that backcountry camping permits may not be issued at certain times due to bear encounter potential. For instance, NPS won't allow camping at Yellowstone Lake before mid- to late-July many years. Maybe you could go while the bears are still denning. Just plan on being very, very cold. Water might not be open yet.