I am assuming that you are an experienced trail runner as much of the surface is fairly rocky. If my memory is correct, you will be able to open up on the plateaus after the steep elevation gains. I am much more of road runner and laced my hiking boots when I hiked into the mountains.
There is a trail that runs along the river in the valley that you can pick up right in town that we ran on.
I cannot comment on the snow as I was there in July. I know that you did not ask about Annecy, but there is an asphalt trail that runs around the lake which was spectular. We did 5 miles and turned around to go back to our hotel for a total of 10. Much different than trail running in Chamonix, but if you plan to go to the town, it is something to plan for. I am told that it is 42K and they hold a marathon on it.