June is a little early - there is snow and some avalanche warning. Some passes will be difficult to traverse given the cornices. July is much better. August, Sept, and Oct are the best.
To add - the Brazeau Loop is a decent trek although the first 12-14KM is very boring. Last year there were several bridges washed out so if you plan to do this one, book it but keep a contingency plan / trip available.
The Skyline as indicated can be a bit busy. But I found it prettier than Brazeau although it is shorter. Easily a 2 day trek (some run it in a day).
I would recommend the Assiniboine Loop. Start at Sunshine Ski Hill in Banff (take the bus to the top) and hike to and around Mt Assiniboine and back again. I believe it would be about 80 KM with 2/3 rds or more of it above treeline. Lots of Grizzly, however. Bring spray.