With more people collecting unemployment benefits over the past two years, this is an increasingly common question that arises when considering bankruptcy relief.
If you are no longer receiving unemployment benefits, then the debt owed for overpayment may be dischargeable. It will likely be treated as a general unsecured debt to the state. If the overpayment was caused by fraud on the part of the debtor, such as continuing to file for benefits after gaining employment, that is a criminal offense and bankruptcy will not discharge that obligation (and there may be a criminal proceeding commenced as well).
If you continue to receive benefits, then the state may commence a recoupment action and deduct a portion of ongoing benefits until the overpayment is satisfied. Note that recoupment is not subject to the automatic stay and will continue as long as there is an overpayment debt and you are eligible for benefits. This is true regardless of whether you file a bankruptcy petition. The automatic stay prevents creditors from attempting to collect on a past debt. The automatic stay will not, however, prevent recoupment via deduction from future benefits payments.