While I would say that the rental peak is between Feb-April, and that a majority of the leases here are May-May, there are still a number of September-September leases that come up. Unfortunately, as the availability (on every campus) various from year to year, it's difficult to predict and wise to start looking now -- at least to get an idea of what is coming available. You could wait to see what's available later, but it is also possible to find a lease with September starts now. Either way, you're taking a gamble. Though your ideal might be finding a place for the next three years, it could be the case that you take what you can get for this year
Since you're out of province, it might be good to get in touch with landlords about securing somewhere now. Do you know anyone currently on campus that can view apartments for you? If nothing else, and you'd like to wait, contact the bigger companies (I'm thinking Keystone, Varsity, Pandew, Springer, Highpoint, Limestone*, Homestead (Brock Towers), Princess Towers) and ask what the typical availability is later in the summer. They likely won't make any promises, but it'll at least give you a general idea of how many late-summer leases typically come available.
I also found during my time here that many landlords were on a program with Queen's (http://community.housing.queensu.ca/landlord-contract-program/) and as such these landlords tend to have earlier requirements for their tenants to renew their lease, and no option for month-to-month contracts. This is a factor that might make month-to-month leases (and therefore August or September 1 lease dates) a little harder to come by, and why I would recommend contacting bigger companies. This way you'll have a better idea of which landlords are on this program and which are not. As an out-of-province student, I would just expect the bigger companies would be more accessible, since independent landlords don't always have websites/public phone numbers. Queen's Community Housing is another resource for finding smaller landlords as well (https://listingservice.housing.queensu.ca/index.php/rental/rentalsearch/action/search/).
I highly recommend that you avoid landlords Daphne Dean and Arnie..something. I can't remember his last name. But from rumors and personal experience, they are terrible. They tend to charge a good base fee, and their units are (sometimes) beautiful and well located, but there are often hidden fees like high-utilities, and if they don't like you they can make your time living with them very difficult. I housed with Daphne and ended up paying well above what I'd expected. She charges for unit utilities, as well as joint utilities (i.e. lights in the hallway, washer and dryer -- which you also pay coin to use), some of her units are oil heat and can be upwards of $150-200/month. We didn't have the thermostat in our unit and the other unit refused to turn it down, so we paid fees we hadn't even expected to pay. Just be aware that while there are nice units here, there are also slumlords who own them.
In terms of general difficulty, I don't know if it's any more difficult than it would normally be on a campus to find a house. I never had any problem and moved every year during my undergrad. Having said that, I'm speaking as an in-province student and I think the difficulty comes from being an out-of-province student.
I'm not sure when Queen's gives you access to your NetID/Queen's email, but if you're able to get on the Facebook pages, the housing page would be a good resource to scope out if any students aren't intending to renew their lease and are on a month-to-month agreement. If you're interested in roommates, that page is also a good place to find them. I also wouldn't knock making an add on Kijiji or surfing ads on Kijiji for September leases.
Despite all of this I don't know if there is one, specific "good time" for you to come out and look. Leases fluctuate, and the landlords who are on contract will have likely leased their places for Sept-Sept by the end of May, barring any issues like quality/cost/etc. The leases that run month-to-month will come up as they come up. I would say that you should come when you feel that you need to have a lease signed for. If it would stress you out to wait, come sooner rather than later. If not, come when you're ready and it makes sense.
I know this is a lot, but I'm hoping it was helpful to you. Good luck!
*I know Limestone also works as property managers for smaller landlords, so this might be a good resource for you.