There is no one best time.
The backpacking season is short this far north and the season changes very quickly. Similarly there is a very large variance of conditions depending on where you go. A trip to the Kenai peninsula in September is very different to Gates of the Arctic NP at the same time. Alaska is wonderful late August/early September, the tundra colors blaze with orange, golds, reds and every imaginable variation of these.
Typically there are fewer bugs later in the year, with late May, June and early July the worst. May/June brings forth cute baby critters, blooming wildflowers, and the vibrancy of spring. It’s largely a matter of deciding what works best for you, what you want to see, and how best to accommodate that.
Snow can fall at any time in of the year, and weather is completely unpredictable. The days are longest during the latter weeks of June and first part of July, but those long glorious evenings last all the way up to late August.
