Thucydides' recording of Pericles' Funeral Oration is mostly a big long list of things Athenians think make them better than everyone else. Granted, the "everyone else" in this situation is Sparta, but it still gives a good idea of the values and ideals of the time (or at least those that Thucydides attributed to them).

Herodotus (author of "The Histories") is a great source on what's Greek and what's not, since he records the causes (often rather legendary) of the Persian wars, so he discusses the histories and personalities of Greeks and non-Greeks. Sometimes you have to read between the lines a bit, but for example, in book 1, the Lydian king Croesus, who ruled over a kingdom in central-western Turkey, is fascinated by Greek customs. He solicits advice from the Athenian wise man and lawgiver Solon, and he sends envoys away to many Greek oracles to figure out what to do about those Persians. The trouble is, Croesus consistently misinterprets the cryptic oracular responses, and the bottom line is that he can't understand them correctly the way that a Greek leader could.

For contrast, Herodotus also tells of the Delphic oracle advising the Athenians to build a wooden wall when the Persians were approaching. Some of the Athenians take refuge on the wooden-walled acropolis, but Themistocles correctly interpreted the oracle to mean that he should attack the Persians with a wooden wall of warships.

"Greekness" is closely tied in with language (J. G. Droysen's old but influential "Geschichte des Hellenismus") and speaking Greek, and this works on a literal level of course, but also on a more metaphorical level, according to the way that Herodotus frames these important episodes.

For them, civilized and Greek were the same thing. As often said "Barbarian" has is root in Barbar, the Greek version of our "bah bah" of a sheep. Any language that wasn't Greek sounded like the baying of sheep. Over time, with greater contact with the Romans, Persians, etc. this prejudice began to fall away, but it was still firm around 440 BCE. It wasn't until the Pyrrhic Wars that the Greeks thought the Romans civilized and allowed them to particpate in the Olympics. Hell, there was still a lot of debate on whether or not Makedon and Epirus were civilized until Alexander and a lot of snobbery against them after, saying that they weren't "true greeks". If you were to take the opinion of the snobbiest greek you could find in 440, a Lady Violet of Athens if you would, she would tell you that only Attica, Boetia, the Pelopennese, and the nearby islands were truly "civilized"