In 2004, I was convicted of criminal contempt in a non violent felony, how do I go about getting my rights to own a gun?

Your best option would be to have the felony expunged(removed) from your record. Once that is done you can have your rights to own a gun restored. You can request that your right to own a gun be allowed without the expungement but that is the best way to clear your record and increase your chances of being successful in your request. I highly recommend that you find a local attorney to help you through this process.