When are you eligible for a restricted driving permit in illinois for a Dui conviction?

Your question contains insufficient information to definitively answer and you should seek counsel. Some 70% of petitions for RDPs in Illinois by my estimate are denied, and an attorney will give you much better chances.
For now, I'll assume you received and completed supervision for the 1st DUI, and a conviction for the 2nd DUI. You would then face a one-year revocation and theoretically would be eligible to apply for an RDP 90 days after the revocation begins. There are other factors, and I am speaking only generally.