Antigone's father, Oedipus, placed a curse on his entire family. The story is told in the first play of the triology, Oedipus Rex (Oedipus the King).
Briefly, when Oedipus was king of Thebes, he vowed to find the person who was causing a plague on his land. He was told he needed to find the person who killed his father and married his mother. Through the course of the play, Oedipus comes to realize that he was the son of the late king of Thebes, Laius, and killed Laius on his way to Thebes. Oedipus then married Jocasta, Laius' widow, thereby marrying his mother. Because of the murder and incest, a curse was placed upon all of Oedipus' children, including Antigone.
Antigone blames this curse of the tragedies that befall her, including her death sentence from Creon.