To start with though, you must realize that what you are looking for is a property owner who, for a handful of money, will provide a stranger with keys to a property worth many thousands of dollars, hoping for the best. Experienced property owners have likely been hurt financially by tenants in every manner imaginable. They are inherently "wary". They try to protect themselves from future losses every way they can. They tend to prefer to rent to stable people with assets, an income, and a good track record. Many property owners have a handful of judgements given to them by a court of law, that are virtually worthless because: they can't find their former renter; the renter has no income or assets; or the renters debts were cleared through a bankruptcy related to a hardship.
I suggest that you avoid large apartment buildings and complexes. I think no matter what you do, you'll be wasting time and if a background check is required, you'll be wasting background check fees and/or application fees. A duplex, triplex, or 4-plex owned by a small investor is a better bet because many don't bother messing with credit reports. If renting a room in a large home with shared common areas like a bathroom, living room, and a kitchen would work, you might be able to find such a place in the Uptown area or near the U of M. Those same areas would be good places to look for something more than a room. Actually, properties near any college might be a good bet...especially if you can move in June since they often lose their student renters at that time.
It's best to be honest about your situation, but I wouldn't mention your credit unless they do. If there's a fee charged to you for a "credit check" and yours is really bad, that's when you should start explaining, instead of writing checks and hoping for the best. Start out by saying that you understand their "situation" (see paragraph #1 above). If you are employed, say so. Tell them what you make in a month and explain that you'll have no problem paying the rent and promise to do it on time. Also, offer to keep the place neat, tidy, and clean. Explain that you won't disturb other tenants. If you don't smoke and/or don't take drugs....say so. Explain that you'll leave the place in great condition and do all of this while telling the truth and while looking directly at the top of their nose while speaking. No matter what happens, don't beg, don't cry, don't argue, and don't lose your cool. If they say "no", leave them with your name and contact info and ask them politely to reconsider and to contact you if they should happen to change their mind.
If you have pets like a dog or cat, you are going to have a much more difficult time finding a place to rent. You might want to work on a "plan B" and find them a home so you can find one. Finding one that allows pets or smokers will be much harder. Perhaps it's time to quit smoking, if you do. If you don't smoke, don't have pets, don't take drugs, don't make noise, and don't drink to excess...be sure they know that...and find a way to tell them that early in the conversation.
I think the best place for you to look for places might be on bulletin boards in grocery stores, etc. and in small community newspapers. The Minnesota Daily is a good source for rentals near the U of M. City Pages also has rental ads. The Tribune might work for you, but it certainly would not be my first choice. I think I'd start with Craigslist. Actually, you can place an ad on craigslist for free.