I was just by there the other day. Road is still closed due to snow.
Lots of camping options up this canyon. A paid one not too far out of Alpine (right where the road first crosses the river). Then a bunch of good free camping up further. There is a nice large open field by the Squaw creek area.
Snaggletooth can be scouted from the road on the way up the canyon (hardest section of this river). Way up further on the grey's you can run into wood issues.
Check out the sections here:
Greys River - River Brain
most of the main river features have been noted and plotted.