The distinguishing feature of Greek tragedy that relates it to Soyinka's play Death and the King's Horseman is that the hero suffers death (or some other horrific consequence like blinding and exile) as a result of a tragic flaw in his character that leads to a fatal mistake or decision resulting in vastly horrible consequences, including the hero's own tragic end. In Soyinka's play, this scenario plays out exactly as in Greek theatre.
Elesin's tragic flaw is his womanizing vanity:
ELESIN: Come then. this market is my roost. When I come among the women I am a chicken with a hundred mothers. I become a monarch whose palace is built with tenderness and beauty.
This flaw, his vain enjoyment of his prowess with women and his thirst for them,
ELESIN: ... the smell of their flesh, their sweat, the smell of indigo on their cloth, this is the last air I wish to breathe ...
is what causes him to make his fatal mistake: He becomes distracted from his purpose in the love embrace of a woman whom he has never seen before on the night of his ritual sacrifice for the king's death.
The horrible consequences of his fatal mistake are that he fails to enact his ritual soon enough; he is arrested so that Pilking, a British civil representative in Nigeria, is not compromised before the visiting Prince of England; his eldest son, who is disgusted with his father's failure, ritualistically offers his own life for the dead king so the balance of the universe may be maintained according to custom.
The final consequences of Elesin's fatal mistake is that, even though chained and in prison, he manages to perform his ritual sacrifice after Iyaloja and his new bride, pregnant with Elesin's child, bring his son's body to show him the fatal consequence of his failure.