Hopefully by then most of the lakes will be ice free but you never know. I was at a lake at 4000' last weekend and it was still mostly frozen and the water was probably 38 degrees. It will definitely depend on what elevation you're at, the end of June is actually still early for most alpine lake fishing in a typical year. For flies bring small leeches and soft hackle wet flies. Midges, Parachute Adams and ants are all good dries to have.
I am no expert, but I would expect most of the higher lakes (above 4000-5000') to be pretty frozen still. If you stay below 4000', you should probably be fine depending on the area and sun exposure. Last weekend driving over Snoqualmie Pass, it looked like most the snow was gone in the 3000'-4000' range. Fish are hungry after ice out.
Friend of mine was out hiking Talapus and Olallie last weekend and said he encountered a couple of feet of snow in the trees. Might see a lot of snow still on the ground unless you are on the east side of the mountains.
From looking at the reports on WTA, it looks like most places under 4000' are snowfree. In a week or so that will move up. You should be alright in terms of snowpack, the water is gonna be pretty cold, might not be too productive.
Whether or not you'll find snow is not so much dependent on altitude as it is exposure. A 4,000 foot lake in full sun will likely be completely ice-free while another one at the same elevation but on the north side of a steep ridge or cirque, and thus in shade, will likely still be iced.