Fellow Podder from Calgary Alberta here. Banff is my play ground.If you are looking for full hook ups then Tunnel Mt. Main (trailer or Village 2) is the only place in Banff. It is set up like a KOA and certain sections don't have very much shade or privacy for that matter. You can view a pic of most sites on-line when booking. (https://www.pc.gc.ca/en/pn-np/ab/banff/activ/camping) It is up hill from the main town site but and easy 5 min drive or a bus ride is available and bike paths as well. If you don't need or want the full hook ups then I suggest Two Jack Lake side. It's on the other side of the Trans Canada HWY from the town site but as it says some sites back on to the lake and a few back on to a creek. Half the sites are walk in tent and the other half are trailers under 25ft. Be ready to book as soon as possible (Jan 15 th for the 2018 seasons) pick two or three spots in each camp ground you are interested in because they do fill up quick. If you want to get off the beaten path a few hidden gems out side the national parks are Kananaskis Lakes (excellent camp grounds) or my fav. Spray Lakes. It's behind Canmore and up the Spray Lakes/ Smith Dorrien Trial. Which has a steep, narrow gravel climb into the sub alpine behind Canmore. But you will be camping right on the lake with maybe one or two neighbors in view. This is as close to backcountry camping as you can get in the Alberta mountains.
Weather wise, you could get all four seasons in a day during that time of year. Mid August I was back packing behind Lake Louise ski hill and woke up to a foot of snow. I'd recommend bring toques, gloves a warm sweater and a good rain shell would cover it. No snow will stick around for long if it dose happened. And the temp can jump around from 0 degrees c at night to + 20 in the day.
Happy Camping and let me know if you have any other questions along the way