How did Cleopatra change Rome's story from republic to empire?

Cleopatra had nothing to do with changing Rome's history from republic to empire. Rome was already an empire under the republican form of government. Because Cleopatra was the last queen of ancient Egypt and was defeated by Rome, her country became another Roman province -- an addition to Rome's empire. What she did do, was to help bring down the republic and enable Octavian to establish the principate.