One of the main features of a Greek hero in its original form was that the hero was a demi-god where one parent was a god, the other human. This was not always strictly the case, as Odysseus is a good example, but a connection to a God whether as a descendant or as a protector was usually celebrated.
A classical Greek hero also had to die while in some noble pursuit as that was an important requirement in the formation of a hero cult, one of the main facets of ancient Greek religion. Obviously a live hero cannot be worshipped as easily as a dead one.
A hero was also usually not perfect and possessed some sort of flaw, in the case of Odysseus his was likely his impatience and his pride which got him into serious trouble several times. This flaw sometimes led to the hero's death but was always a prominent flaw.