June should have about the best summer weather. If it's near the end of June, the Taste of Chicago starts for about 10 days--dozens of restaurants have small samples of their food along with entertainment in Grant Park.
There's Second City in Old Town--the famous comedy club. Blues clubs everywhere. World class museums--Art Institute, Field Museum of Natural History, shedd Aquarium. On Oak St. crossing North Michigan Ave, you'll find Prada, Tod's,
Kate Spade, Hermes, Jil Sander, plus local boutiques. Marc Jacobs just opened a store but it's not near Mich Ave.
In the loop (downtown) along the lakefront is Millenium Park where you can see the newest "landmark"--a steel sculpture nicknamed the Bean. It reflects the city skyline and everyone takes a pic of themselves standing in front of it with the city reflected behind you.
If you're there the second full weekend in June there's the juried Old Town Art Fair--very broad selection of high quality, but mostly affordable art.
I think Twiggers is correct about the Oprah show--not taping in June and it's nearly impossible to get a ticket anyway.
You'll have a great time!