Where are you moving from? How old are your children? Are you talking about living within the City limits (some jobs have residency requirements) or just near the city, in order to have a reasonable commute to a job downtown?
The "city," including suburban areas, stretches more than an hour's drive from north to south, east to west....and cost of apartments/availablility vary tremendously. The City of Chicago has a magnet school program, and many of them have good extracurricular programs...my nieces were involved in track at their high school...Here's a recent discussion thread where another poster asked about moving to Chicago, and the responses and resources listed there might be helpful to you: community.greatschools.net/q-and-a/169992/Hi-Everyone-We-are-relocating-from-the-UK-and-need-advice-on-neighbourhoods-and-schools-in-chicago-Anyone?cpage=1#