Here are my 2 cents. We did a unexpected trip through Banff and camped near Lake Louise. It was a beautiful site, about 2-3 miles from the lake. We went at the beginning of August and had no problem getting a site. The weather was 70's in the day and low 50's at night.

I love that area and am envious of you.

Just wanted to wish you a great trip.

I've camped/hiked/climbed around Banff and Jasper several times. I would recommend securing reservations ahead of time since you will be there at peak season. If you don't you may end up wasting time having to drive to more remote campgrounds that are not as handy to places you might want to visit. I personally like the Lake Minnewanka campground outside of Banff for comfort and convenience if you get a stretch of rainy weather (you can reserve at http://www.pc.gc.ca/pn-np/ab/banff/visit/visit9b_E.asp)
Beside lovely tent sites it is near a lake you can canoe, mountain bike and hiking trails and just minutes from Banff townsite (if the weather gets really rank, spend a day at the 3 museums in town, the Rockies museum, the White and the Indian Museum). Minnewanka campground has toilet rooms with indoor plumbing and covered picnic rooms with concrete shelves for campstoves. It always rains at least a couple of days a week in August up there so this is a good place to stay in such weather. There are no showers but you can drive up to the Sulphur Mountain Hot Springs spa above town to soak in hot water and take a great shower -- heavenly after a long day's hike.

Forty Mile creek is a nice hike that starts at the downhill ski area above Banff -- when I was there 30 years ago we found signs of recent grizzly activity (and a bear ripped the door off the van parked next to ours to get food the owners had left in it!)
For more rustic accommodations, the Climber's campground by the Columbia Ice Field (outhouses and one water tap) has great views of the mountains and is near some gorgeous day hikes -- Beauty Creek will blow you away with it's waterfalls and you can hike around the icefield. (http://realtravel.com/jasper_national_park-journals-j2532338.html)
Don't go up onto the icefield without companions and mountaineering gear, though. There are hidden crevasses and you can get in trouble.

Don't forget to register at the sign-in boxes before embarking on any hike. it used to be that if you didn't sign out and back in and ended up needing a rescue, the government would charge you for it (and helicopters are $3,000 per hour!)

As to hiking alone, I would not recommend it except in the most well-frequented and moderate trails like the Tea House trail at Lake Loiuse. It's rugged terrain and there can be drastic weather changes. it's easy to meet up with fellow travelers, though, to join up with. One really good place is the YWCA hostel/Inn in Banff. A very inexpensive place to stay with private rooms and dormitories, by the way, but even if you are camping, it has an inexpensive restaurant and a small bar and lounge where you can meet other folks. There are some day-hiking trails that leave from just behind the Inn and it's right at the end of town.

Have a great trip -- it's one of my favorite places to hike in the world (except for the darned rain!)