Γεια σου Ιακοβε
It is not really important how you will spell your surname, since it is not a Greek one.
You can apply the Greek masculine ending -ης and spell it as "Μόρης" οr "Μόρρης". In this case you will change the ending according to the Greek inlination of nouns.
Nominative: o κύριος Μόρης
Genitive: του κυρίου Μόρη
Accusative: τον κύριο Μόρη
Vocative: κύριε Μόρη
An other option is just to write it phonetically in Greek: "Μόρις" or "Μόρρις"
In this case, you do not have to follow the Greek inclination.
Nominative: ο κύριος Μόρις
Genitive: του κυρίου Μόρις
Accusative: τον κύριο Μόρις
Vocative: κύριε Μόρις
It is up to you which one you will choose. If you live in Greece permanently, you speak fluently and you feel Greek, you can choose the Greek version.