How did personal relationships between Marc Antony and Octavian and between Marc Antony and Cleopatra affect the history of Rome?

Marc Antony and Octavian were allies in the Second Triumvirate (three-man alliance, the third man was Lepidus) which fought a civil war against the assassins of Julius Caesar, who called themselves libertores.
After winning this war the two men shared power. Octavian took charge of Rome's western territories and Mark Antony took charge of the eastern ones. The relationship between the two men, who did not like each other, turned bad. A propaganda war, with the two men slandering each other, started. Octavian then sought war against Marc Antony. He used Cleopatra VI of Egypt as an excuse to fight Marc Antony. Octavian wanted war because this was a contest over who would become the sole ruler of Rome and her empire.

Marc Antony became the lover of Cleopatra. He had three children with her. He gave kingdoms in the eastern territories to these children. Alexander Helios was made king of Armenia, Media (in present day north-western Iran and south-eastern Turkey) and Parthia (in present day north-eastern Iran). Most of Media and Parthia were not under Roman control. His twin sister Cleopatra Selene II received Cyrenaica (eastern Libya). Ptolemy Philadelphus received Syria and Cilicia (on the Turkish coast next to Syria). Cleopatra, she was proclaimed Queen of Kings and Queen of Egypt and was made the ruler of Egypt together with Caesarion (the son of Cleopatra by Julius Caesar), who became King of Kings and King of Egypt. Caesarion was declared legitimate son and heir of Julius Caesar. Octavian perceived this as a threat to his bid for power, whose legitimacy was based on him being the adoptive son of Caesar. This had made him popular and had secured the loyalty of many of the Roman legions to him. This is when the propaganda war started.

The acrimony between Octavian and Marc Antony divided the Romans. Many senators and the consuls of the year left Rome to join Marc Antony. Important supporters of Antony defected to Octavian. Octavian accused Marc Antony anti-Roman sentiments. The senators were not bothered about Marc Antony partitioning his territories among his children, but were incensed by the fact that he wanted to be buried in Alexandria of Egypt instead of Rome. Octavian blamed this on Cleopatra and had the senate declare war against her. This was an excuse for fighting against Marc Antony as Octavian knew that he would fight to help Cleopatra. Octavian defeated the Egyptian forces of Cleopatra and Marc Antony's troops. Antony and Cleopatra both committed suicide.

After winning this civil war Octavian brought the Roman Republic to an end, established his absolute personal ruler, became the first Roman emperor and thus started Rome's 503-year period of absolute rule by emperors. He also assumed the honorary title of Augustus.