Chicago to GSM is almost exactly 600 miles and is right at the limit we recommend for a single day, you are looking at about 12 hours. May 4 is a Monday, and you need to get out of Chicago by dawn to avoid the morning rush hour traffic. Doable? Yes, but don't try to do it day after day. I would probably spend the night in Gatlinburg.

Skyline Drive? Note that it will take you about 5 hours to drive from Waynesboro to Front Royal on it. Front Royal to NYC is 6 hours via I-81/I-78.

If you are looking at taking the Blue Ridge Parkway from GSM to the Skyline Drive, this is a multiple day trip - 2 to 3 days. Otherwise, GSM to Waynesboro is about 7 hours via I-81.