Capitalized:
* When referring to the people, language, or culture: "The Greeks have a rich history." "Learning Greek is challenging." "Greek mythology is fascinating."
* As part of a proper noun: "The Greek Orthodox Church." "The Greek island of Santorini."
Not Capitalized:
* When used as an adjective: "Greek yogurt." "Greek mythology."
* When referring to the language in a general sense: "He speaks fluent Greek."
General Rule: If you're unsure, it's generally safe to err on the side of capitalization. However, if the context makes it clear that you're referring to the language or culture in a general sense, then lowercase is acceptable.