Early in the final book of Homer's Iliad, the god Apollo criticizes the other gods for allowing Achilles to defile Hector's corpse by dragging it behind his chariot. Apollo argues that Hector had always honored the gods with sacrifice and now his corpse is being defiled. Apollo goes on to say that Achilles is behaving like a wild animal and that
"Achilles is as devoid of pity, and of the shame that benefits men, urging restraint." (A.S. Kline translation)
Thus, it appears that the sort of shame Apollo describes is that which restrains a person from behaving in a manner that would be displeasing to the gods. By defiling Hector's corpse in this brutal way, Achilles is acting in an unrestrained manner. Such disregard for common decency is probably what the Greeks would label as hubris. In Greek mythology, committing an act of hubris will almost always result in a person being severely punished by the gods.
The sort of shame that Achilles should be exhibiting is an attitude of respect for the notion that all people deserve a proper burial after they die, even a person's enemies (compare Sophocles' Antigone).