No, you cannot combine the refund from one state with the balance due owed to another state and report the net amount. Each one must be handled on their own.
The balance due that you had to pay to Illinois will be included in the Taxes You Paid section of Schedule A Itemized Deductions. However, this section has a maximum limitation of $10,000 (or $5000 if Married Filing Separate). Plus, you must be using Itemized Deductions instead of the standard deduction to take advantage of the deduction.