Pretty easily. If your conviction was in California, you should probably (but not necessarily) hire a lawyer for the procedure.
What you want is to file a petition under Penal Code section 1203.4 as soon as your probation ends. If you have a good record on probation, they might agree to terminate your probation immediately. In the petition, you must show that the crime of which you were convicted is a "wobbler "(chargable as either a felony or a misdemeanor), that you have completed probation without having been sent to state prison, and that you currently have no other criminal charges pending. Then you ask the court to reduce the felony to a misdemeanor, then ask that you be permitted to withdraw your guilty plea and enter a new, not guilty plea. Then, the court dismisses the charges under PC §1203.4.
Although this result does not relieve you of all of the penalties and disabilities which you incurred as a result of your felony convicton, your convition will be expunged and prospective employers will not see any conviction - neither felony nor misdemeanor. If you were to retain a lawyer for this, you should expect to pay somewhere between $350 and $750, plus a $60.00 filing fee.