Okay, first of all, as it has been said, what state? And then, how long ago did all this occur? What you got was a letter informing you that you were being investigated for unemployment fraud, that according to what they'd found, you had apparently received benefits for which you were not eligible. This was either determined to be a fraud or non fraud overpayment.
I'd suspect, depending upon the state, that you probably did not receive these benefits after being approved and then later had a claim overturned by decision, which would have resulted in a "non-fraud" overpayment. Somehow or another, you got benefits for which they do not believe you to be eligible. Depending on the state, they may have assessed penalties along with the actual amount that you were determined to have received in error.
So you got a letter (again, how long ago?) saying that you have been determined guilty of unemployment fraud and that you have been overpaid and are going to have to repay xxxxx amount of money. Usually any further appeals are discussed at this point, and a strict time limit to make such an appeal is stated.
Okay, were you given a number to call to discuss repayments? You may disagree with the decision that you in fact committed the fraud or that you were overpaid, but what has happened is NOT a criminal charge of fraud, and "clearing your name" is irrelevant. The unemployment system is closed, and that you have been charged with unemployment fraud and owe the department xxxx amount of money is not going to show up on any criminal record, is not something you'd be supposed to report on applications ( since you have not been prosecuted for anything by the criminal justice syste, this is an agency) and is not going to affect your credit report. Until later, when they begin trying to get that money back.
The thing is, repayment of the money you received is what they're interested in, though if they choose to do so, they
could, actually recommend you for criminal prosecution. And what they're most likely to do, depending on what state you are in, is take steps to recoup the money. Such steps as that if you are now working again, you may find your wages garnished. If you receive a state income tax refund, they may confiscate it. They may keep you from renewal of your state licenses of any kind that you have until the money is repaid.
My suggestion, in MOST states, is that you call the number on the decision letter you have received last, ask if it is too late for another appeal, and talk to them about setting up a repayment plan, or possibly, if you are not able to repay the money due to financial hardship, in some states you can request a waiver of the overpayment, though if this is a fraud overpayment, that's less likely. But you have to tell us more, and it is obvious you are not really understanding what went on during this investigation and appeal process.

