How did Octavian gain power over Rome? A question asked on Quora.com and my answer is expanded here.
Octavian was already powerful, by accident of birth, by becoming the inheritor of Caesar's estate and status, and by forming an illegal army from Caesar's veterans which brought him to the attention of the Senate. They wanted him punished, but Cicero persuaded them the youngster would be useful.
At the age of nineteen, on my own initiative and at my own expense, I raised an army by means of which I restored liberty to the republic, which had been oppressed by the tyranny of a faction. For which service the Senate, with complimentary resolutions, enrolled me in its order, in the consulship of Gaius Pansa and Aulus Hirtius, giving me at the same time consular precedence in voting; it also gave me the imperium. As propraetor it ordered me, along with the consuls, "to see that the republic suffered no harm." - Res Gestae (Augustus)
Octavian rose to dominance by manipulation of popularity, by acting against Marc Antony whose association with Egypt made him a traitor in the eyes of the Senate and People of Rome, by defeating the Romano-Egyptian threat to Rome, by annexing Egypt by conquest, and by returning the supreme power he had been awarded to defeat Cleopatra to the Roman state, then by a series of intense negotiations with the Senate.
But please realise Augustus was not powerful in one title. He held a number of magistracies, priesthoods, and privileges that together added up to some serious influence. He was effectively manager of republican government, but never an emperor despite the popularity of that idea. He retained Egypt as his own personal province where the Senate could not legally go, he had direct influence over regions with military garrisons thanks to his senior imperium (right to command an army), but a third of the empire remained under senatorial oversight.
For ten years in succession I was one of the triumvirs for the re-establishment of the constitution. To the day of writing this I have been princeps senatus for forty years. I have been pontifex maximus, augur, a member of the fifteen commissioners for performing sacred rites, one of the seven for sacred feasts, an arval brother, a sodalis Titius, a fetial priest. - Res Gestae (Augustus)
Also please note that descriptions of Augustus changed over time. He was regarded as the saviour of the Republic during his day (which to be fair he was only too keen to encourage), but five hundred years later he was being described as an absolute monarch who abolished the aristocracy. Little wonder the Middle Ages decided to call him an Emperor however wrong that was.
May it be my privilege to establish the State in a firm and secure position, and reap from that act the fruit that I desire; but only if I may be called the author of the best possible government, and bear with me the hope when I die that the foundations which I have laid for the State will remain unshaken. - Augustus
Consul - One of two annually elected senior magistrates of Rome, also originally military commanders of a legion each.
Imperator - Victorious General, described by Cassius Dio and Varro.
Princeps Senatus - First Senator
Propraetor - Former leader
Supreme Power - Not defined by anyone but indicates the powers of a Dictator without the title, which had been abolished by Marc Antony after Caesar's death.
Triumvir - Member of a council but in Octavian's context, a reformer of the state.