You have a couple of avenues available to you. However, you must still qualify on the basis of your income. Your income will be adjusted for any children you have, average medical costs, etc. You might just make too much money or have a higher standard for apartments than you can readily afford, which doesn't automatically qualify you for assistance.
There are two basic types of rental subsidy--project based and voucher/certificate. With a voucher or certificate, you must find your own apartment (ie a landlord willing to accept the voucher or certificate). The housing authority pays the landlord directly, so it's a pretty good deal for them--much more reliable than some tenants. The v/c will be worth a certain amount, and if your apartment is more, you have to pay the difference. If it's less, you don't get to keep the difference. Project based subsidy means that the unit itself is subsidized. If you leave the unit, you lose the subsidy. In project based, you are either placed by the housing authority, or you apply to the building directly yourself. Project based can mean public housing, or it can mean a privately owned building that has a subsidy contract with state or county housing authority.
As far as how to get it: again, there are 2 basic ways. Every county has it's own housing authority, so yours would be Cook County Housing Authority. Unless you're willing to move into Chicago, which has its own housing authority. You can apply directly through them for either program and they can tell you what their waiting lists are for apartments or v/c. Your second option is to contact HUD for their list of privately owned buildings that contain subisdy. These are not public housing, so you have to apply to each building indivually. LaShun Perry at HUD keeps that list, and he can mail it to you. His number is ###-###-####
If you or anyone in your household is on drugs or has a criminal record, you can be denied in the privately owned buildings. You might still be able to get public housing, but they've tightened their rules as well. There's usually a grace period--like if it wasn't a violent crime and it's been 5 years, then you might still be able to get an apartment.
Another caveat is that if you don't have children, aren't elderly or disabled, there can be a limit to how long you can receive the rental subsidy assistance as well. For example, they might give you a two year voucher. The project based assistance is much harder to take away.
Hope that helped. Good luck to you.