You really need to contact the closest Greek embassy or consulate near you, they will likely have a web site with the information you require. Try finding them via this web site:
www.embassyworld.com
As for you not being an EU citizen, I'm non-EU myself and working in an EU country. SOme of the EU countries are pickier than others about non EU citizens working in their borders.
Basically, you probably have a chance at getting work, but expect to see EU citizens given preference over you.
Definitely get in touch with your nearest Greek embassy or consulate before trying to do anything though.
Good luck.