The Greek national identity is very interesting issue. It's also a very delicate one when talking to Greeks about it!

It's a complex issue, but essentially ancient Greeks identified themselves as Hellenes. After the Roman conquest of the lands inhabited by Greeks (which did not match up perfectly to the modern Greek nation state), in the last few centuries BC, the Greeks gradually came to identify themselves as Romans (Romaioi) in the Byzantine period (approximately 5th-15th centuries AD).

As the Byzantine Empire declined and proportion of Greek speakers in the Empire increased, there was a Hellenic revival and many Greeks, especially the educated began to identify as Hellenes again (the revival also featured an increased interested in the literature and history of Ancient Greece). This continued until the rise of Greek nationalism in the 19th by which time most Greeks probably identified as Hellenes.

So I would argue that the Greeks had the beginnings of a national identity when the Hellenic revival started around the 13th century AD.