First, you should consult with an attorney to first see if your prior criminal offense and your record makes you eligible to have your record expunged. Some courts have the paperwork online to file for expungement. If your offense was a misdemeanor you need to wait a year after you complete probation. For a felony, it's three years. So long as it was a first time, qualifying offense (not including traffic offenses), you have no criminal charges since then or presently pending, you may qualify.
You file a written motion asking that your arrest and court case be expunged, make sure you serve a copy of the Motion to the prosecutor's office which prosecuted your case and note how and when the service was made on the motion. The court, if it is inclined to grant your motion, should set it for a hearing.
Most attorneys charge a flat fee to handle an expungement which is very reasonable.