If you want to do it on your own you will need to contact the president or mayor of the community for a civil wedding, or the head priest for a religious wedding. Your birth certificate will need to be translated and an ad put in the local papers so that anyone on the island who has a reason for you not to get married can protest in advance rather then come on like Dustin Hoffman in the Graduate.
One of the best reasons to get married in Greece is that you don't have to travel very far for your honeymoon. You are already there. Also how many people have wished they could bring their friends and family with them on their honeymoon? If they made it to the wedding then they will be there. If you need more information on having your honeymoon in Antiparos,click here.
Getting Married in Greece - the Easy Way
Assuming that you believe that a wedding is an exchange of vows before God, then there is an alternative to going through the red tape of the civil or Orthodox wedding, having it done professionally. This would be a simple exchange of vows, with or without witnesses after having been 'officially' married by a justice of the peace in your country. You find the most beautiful, spiritual spot you can and before of eyes of God and either by yourselves or in the company of friends, exchange your weddingvows. No red tape, no permission needed, no money! Just the two of you in your own special ceremony that you will always remember because it was 100% yours. You