I live in Calgary where I have driven that highway many times and hundreds of times between Field BC and Calgary. The others gave you some pretty accurate info about the highway. The Coquihalia (highway 5) between Hope and Kamloops has some steep grades. Rogers Pass will be downhill eastbound thru several snowsheds that have limited visibility so the speed limit is 70 kms (about 40 mph) so it's no problem. Kicking Horse Pass will be uphill from Field to the Continental Divide where I can't recall any RV being overheated in the hundreds of times I've been on that hill.
As already mentioned by it's well worth the time to go to the Columbia Icefields about a couple of hours north of Lake Louise on highway 93. The exit to the Icefields Parkway (highway 93) is just a little west of the Lake Louise exit but it's easy to spot. A park pass is mandatory to drive the highway so they have a booth at the beginning that will sell you one...
http://www.explorerockies.com/columbia-icefield/ Excellent scenery on that highway too...
Snow in June shouldn't be an issue... I've seen it snow in July and August but it's gone as fast as it came so its rarely an issue for the roads...