How long does the process usually take to evict some one from a rental home in the state of Illinois?

You must
1. Give the tenant proper notice to move (typically 3 days), and then
2. File the eviction with your local court.
3. The court hearing will generally be anywhere between 3 and 5 weeks.

Here is a link to your State Landlord-Tenant laws for more details: http://www.ilga.gov/legislation/ilcs/ilcs2.asp?ChapterID=62 (This is Chapter 765. From there scroll down to "Landlord and Tenant").