Cleopatra thinks Antony is going to be furious with her for abandoning him. Cleopatra feigns death so that Antony would not be furious. But plan goes wrong, Antony tried to kill himself. He died in Cleopatra's arms. Cleopatra kills herself.
This is similar to the story of the boy who cried wolf. Inject a little bit of drama, everything goes wrong. Sometimes I think people during Cleopatra and Antony's time are lacking of rational thought. If only Cleopatra could just face the angry Antony, everything would be okay. If only Antony didn't try to kill himself, he would still be alive living together with his lover.
I think this is a lesson to be learned by everyone. Yes, we would do need an ounce of drama to spice up our life, but do get a grip, okay?