If you are intending to travel this summer, I think your budget is likely to be unrealistic and you're probably very late in planning. With the low CAN $, high US $ and free entrance to national parks in 2017, the parks and surrounding areas are expected to see record numbers of visitors. And prices for things going up - supply and demand...
Hotels, B&Bs and hostels are booking up quickly, and reservable campgrounds are also quickly being booked. Provincial campgrounds will likely be equally as full this summer, and are generally no cheaper (or more expensive) than national park campgrounds. Free camping is not permitted in most places (other than in remote areas of the parks and you still must pay a per night or yearly camping fee) - the most common legal location for free camping is crown land, but it's nearly impossible to access crown land without car (and a non rental car since most rentals forbid going on gravel roads).
Canmore is nice, but it's not cheap and there are no campgrounds in Canmore. The closest would be about 20 minutes away at Bow Valley Provincial Campground (name ??). Think there is a closer campground or two, but there's been talk about those becoming day use only. Hostel Bear is certainly OK, though it's likely to be heavily booked in the summer - it's $35 per night plus tax, slightly cheaper for a weekly rate. The other hostel in town is the Hi/ACC hostel - cheaper for members, but small and probably already heavily booked for weekends. It's also outside of the town centre, so not convenient without a car or bike.
Banff has a number of hostels and Lake Louise has a hostel - rates can be up to $55 per person per night for single bed. Wilderness hostels are cheaper, but you need a car to get there.
Bigger issue will be finding a place to stay. Things will be very, very busy, especially on weekends and the holiday weekends. Without a vehicle, you will be very limited in your ability to go from one campground to another to find space. However you go, you need to book in advance - ASAP - it's not the time or the place where you'll be able to simply find a room or campsite, let alone ones that are really cheap.
Food can be done relatively cheaply if you cook your own meals. Really depends on what you intend to eat and how frugal you can be. Best way to get an idea of food prices is to look at the online supermarket websites (Canadian Superstore is one). Anything in the Rockies will be a bit more expensive. You won't be eating out on anything close to your budget - sandwiches will run $5-10, an average main course/burger will be $8-15 or more.
Others issues are going to be transport and camping equipment. Even basic camping equipment is going to take a chunk out of your budget, and cooking at campsite is probably going to be more of a pain/costly than at a hostel. You can't depend on cooking over a fire - fire bans happen, it rains, you're in a hurry (fires take a long time to get hot enough for cooking), your campsite doesn't have a fire pit/box etc. Also, it's not free - national park campgrounds charge a nightly fee to have a fire, and most provincial and private campgrounds charge for firewood. So you need a stove, cooking pots, utensils, plates etc.
Plus, you must keep a bear/wildlife safe campsite. That means storing all food, liquids and scented items in a car or bear proof box. Most non national parks campgrounds are set up for car campers, so there's nowhere to store such items. You'd need to find campgrounds where a bearbox is provided - usually in tent only areas of national or provincial parks.
At a minimum, you'd need a tent, sleeping bags (warm), sleeping pads, stove, dishes/utensils, cooking pots, probably a tarp, flashlights/headlamps etc.
Transport is going to be another issue - car is not in your budget, and without a car, you'll likely have a challenge getting to/from places, especially on a bare bones budget. I don't think there's public transport to any of the provincial campgrounds. Canmore has a single bus loop and a bus link to Banff. Beyond that, you're looking at Greyhound to Lake Louise (very inconvenient times), and not much else (other than in Banff townsite). Almost all trailheads and other tourist sites require a car or tour to get to, and there's no public transport (within the parks) to Jasper.
One option would be Moose Tours - they are low cost and use the hostel system including wilderness hostels.
However, I think you need to sit down ASAP to figure out whether this trip will work. Assume camping will cost $30 per night, plus additional costs if you plan on having fires. Allocate an average of $40-50 per night, per person for hostels. If you are going to camp, there's the upfront costs of camping gear. Then figure out transport costs - you will have to get to the campgrounds/hostels, to supermarkets, to any place you want to see.
