Best time to go? Mid December, two days after a 6"-8" snowfall in the park, roads plowed, clear skies, bright sun, absolutely no wind, temp about 40 degrees, barely needed a coat, virtually no visitors in the park, apparently scared off by the much heavier snowfall in Flagstaff, at most of the overlooks my footprints were the ONLY footprints in the snow except for small animals, so quiet I could even hear the river far below at a couple of the overlooks.
One of those once in a lifetime events I stumbled into about 20 years ago.
But since that is hard to plan for, I would just plan on going outside the normal tourist season, and preferably during the week.