I have a couple , but it depends on how close you want to be to Chicago and what kind of camping you enjoy. I prefer, in general, non-commercial campgrounds so I don't know any private campgrounds to recommend. Starved Rock State Park is probably you're best option in terms of having a lot activiites to do in the area. Especially if you're a hiker. Unfortunately, it's about 2 hrs southwest of Chicago. The north and northwest suburbs have a couple of options that would be approx. an hour away. They are Illinois Beach State Park, Chain O' Lakes S.P., and Moraine Lake S.P.. I would suggest looking them up online to see what you think. The Illinois Dept. of Natural Resources runs the website you'd want to check out.
http://dnr.state.il.us/lands/landmgt/parks

The closest campground you will find is about 40 minutes NW, (all tollway so very easy), from the downtown/lakeshore

Chicago Northwest KOA
8404 South Union Road
Union, IL 60180

Information: (815) 923-4206
Fax: (815) 923-1440
Email: chicagonwkoa@hotmail.com