I know of no general principle of law requiring landlords to rent to people with criminal backgrounds. If you were a landlord, and you had a choice between two renters, and one of them had a criminal background, and the other didn't, which one would you pick to rent your property to?
Check with an Illinois-licensed lawyer to obtain legal advice. Don't take what I say here as legal advice since I don't hold Illinois licensure. It's just my two cents based on the facts you describe in light of general principles of law.
Keep at it and you will find an appropriate place to live. Good luck to you.
There is no law that prohibits landlords from avoiding convicts as renters in Illinois. Indeed, in this desperation economy, with landlords having to carry so many renters month to month without rent payments, it has become routine in Illinois with a 14 % unemployment rate to do criminal background checks BEFORE renting.
Good for you, though, is that some places still do rent to those of you that have fully paid your debt so society. Good luck and keep searching.