Essentially, Augustus gained power in a piecemeal fashion. What I mean by that is he did it gradually in a way that eludes to the fact that he had no 'grand strategy' for becoming emperor rather it was an organic process.

This is not to say he had no designs on becoming emperor; he clearly did but was astute enough that doing it openly, in a manner similar to his adoptive father Ceasar, would result in a similar fate.

The way he became the most powerful man in Rome whilst appearing to have gained all his power legitimately through the Senate is was, as I said, to do it gradually. Upon defeating Mark Antony after the battle of Actium in 31 BC and returning to Rome, Augustus offered to resign all the power he was given upon the creation of the second triumvirate. In what is referred to as the "First Settlement" (27 BC) the Senate 'begged' Augustus to retain his powers and awarded him proconsular imperium and consecutive consulships.

This essentially meant he was the most powerful man in Rome; he had divided the provinces into two categories Senatorial and Imperial. Only Augustus could control the Imperial provinces (they were the richest and had the most legions) meaning he controlled the majority of the wealth and military might of Rome, all whilst maintaining the idea of the 'Princeps' the concept that he was first among equals. The most powerful, but with power that had been awarded to him by the Senate.

In 23 BC Augustus further increased his power via the "Second Settlement". Essentially the Senate awarded him tribunicia potestas (the powers of the tribute of the plebs) and Imperium Maius (literally 'more power' than any other man). With these two powers, he had the ability to pass and veto any legislation, call meetings of the Senate, and could override the command of any General, Senator or Official in the Roman empire.

As was the case of the "First Settlement" the "Second Settlement" had the appearance of the Senate awarding Augustus power when in reality he himself had engineered these settlements. This was how he maintained the facade of the principate; the notion that this was not a autocracy but still a republic. By using the Senate to gain his power, no-one could accuse Augustus of being a king as he had gradually and, by all appearances, legitimately gained his power. Furthermore, he absolutely avoided direct succession. Rather, he groomed certain individuals to take his place that was not a blood relative. Again, this way no one could accuse him of being a King and creating a dynasty; even though he did.

Combine this with his moral and religious reform which painted Augustus as the restorer of the religious tradition and moral value that Rome had lost in the past decades and there you have it. Augustus was seen as restoring the Republic to its pre-civil war glory all whilst drastically changing the political structure of said republic into an Empire. He did it so successfully, partly because his strategy was constantly evolving, that no-one noticed. And if they did, well they had no power to do anything about it.

This is a pretty brief explanation of how Augustus gained his powers. There may be some errors- I'm not a classicist. However, I do hope it is helpful and I implore you to do more research. Don't just take my word for it; read books, watch documentaries etc, to develop your understanding.