One example of Greek culture seen in Sophocles' Oedipus the King is the consultation of oracles. Delphi was a very real place and real Greeks went there in the hopes of consulting Apollo's oracle. A response from the Delphic oracle started the famous Socrates on his quest to discover what it meant to be truly wise.
Another example of Greek culture seen in Sophocles' play is connected with the plague that affects the Thebans. Many Greeks believed that a person who had committed a crime such as murder was "polluted" in a religious/ritual sense. This ritual pollution was known as miasma. Therefore, the notion that the mere presence of Laius' killer within Thebes could cause disfavor from the gods was a very real fear in ancient Greece. As in the case of Oedipus at the end of the play, a person who was determined to be polluted in a religious sense could be exiled from a city-state. We even hear of instances in ancient Greece where objects such as statues were regarded as polluted and therefore were cast outside the boundaries of a particular city or island.
Finally, Sophocles' play appears to have been first put on in Athens around the time when the Athenians themselves were reeling from the ravages of a terrible plague that may have wiped out as much as a third of their population. Thus, Sophocles' audience may have identified closely with the mythical Thebans' situation:
For our city, as you yourself can see,
is badly shaken—she cannot raise her head
above the depths of so much surging death.
Disease infects fruit blossoms in our land,
disease infects our herds of grazing cattle,
makes women in labour lose their children.
And deadly pestilence, that fiery god,
swoops down to blast the city, emptying
the House of Cadmus, and fills black Hades
with groans and howls. (Ian Johnston translation)