About the middle of May to the third week of June. May would be the best time because the leaves haven't unfurled yet and you'll have a better chance of seeing them. But it depends on the snow too. We've had a great winter up here this year so I don't think you'll have to worry about getting in.
There are more bears out during June but they'll be harder to spot. I don't know if you've ever been up here but it is THICK in some places.
I'd take a day around the first of May and drive down, find the trail, see what the snow conditions are and look the place over. You can make it down and back to Anchorage in a day easy.
My wife and I only hunted it once and the snow was still pretty deep. That was the third week in June 2011. We had a lot of snow that year. I'm surprised we didn't see more bears. There were TONS of tracks everywhere.