If so how much would something like that cost
No you cannot. The only exception that I can imagine, and it would be a very rare one, is if there were new evidence of a constitutional violation that you could not reasonably have discovered until now. That might perhaps warrant an out-of-time post-conviction petition. But such a petition would be a very, very long shot, very expensive, and with little hope of success. Probably a waste of a lot of money. Your best shot at undoing this is to let a lot of years go by, like maybe twenty, live it down, and then ask the governor for a pardon.
Shy of a statutory available Governer's Pardon you cannot reonen that conviction absent of any manifest errors in the case. Even the Pardon's grant is extremely hard to get and very costly.
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Criminal Defense Attorney
A five-year old conviction cannot be re-opened. Only if you were still incarcerated based on that conviction would you have the possibility of a habeas corpus petition in federal court alleging violation of a constitutional right. A petition for excecutive clemency/governor's pardon is your only option.