How Augustus acquired all of his powers has to be seen in the context of the period. The Roman Republic was falling apart and had experienced either constant civil unrest or open civil wars for the past half century. By the time Augustus emerged as the sole surviving power (having outlasted Brutus/Cassius and the 'Republicans', Lepidus, and Antonius) the Romans were pretty exhausted with the constant civil wars, taxation to pay for those wars, proscriptions, and confiscation of land to reward Augustus' soldiers.

Augustus, having defeated anyone else of any stature, offered to the Roman state peace. He was extremely clever in how he managed to accrue power by always using Republican precedents or language. Some of his power was purely personal, his auctoritas at this stage was unparalleled and his opinion carried the weight of law.

Backing that up, he acquired formal legal powers through the Republican framework. All of these were granted by the senate - almost certainly stage-managed behind the scenes by him or his close adherents.

• He was given the tribunician powers making him the defender of all the plebs - but that power also gave him the right of veto and the ability to introduce laws before the Plebeian Assembly. This wasn't a power he threw about freely, the tribunes had contributed to the fractious politics of the Late Republic and carried a 'rabble-rouser' connotation to the elites which Augustus now courted. The veto was something he could deploy, a political 'nuclear option', but most of the time his simple personal authority would be more than persuasive.
• Using the precedence set by Pomepius, he was made governor of several important provinces which he then controlled through legates (deputies he personally selected and under his personal authority). As governor he then possessed imperium, the right to legally command troops and his provinces (as opposed to the remaining 'senatorial' provinces) contained the bulk of the legions. So no senator could command a large army to threaten Augustus' position and he either appointed very trusted senators (who wouldn't turn on him) or knights who were of too low stature to try to seize power.

These were the main struts of his power. He ultimately controlled the armies of Rome, possessed a veto over the senate. But it was his wealth, his patronage, his auctoritas that enabled him to rule with the support of a decimated and relieved senate & equestrian class. To maintain his high prestige, he insured that no other senator would be allowed to gain similar glory. So he began limiting and eventually restricting triumphs to him or his heirs. He took the title of imperator, an appellation given by soldiers to their successful general, and made it a title restricted to him (which would pass down to us as 'emperor'). He was given the title of 'Augustus' which carried a religious connotation to raise his standing above that of mortal men (it would become a title signifying senior emperor by the Late Empire). He played up the deification of his adopted father Julius Caesar so that he was 'son of a god'.

Augustus tried to designate an heir but unfortunately he outlived all of them. After the last one died he was forced to turn to his stepson Tiberius who he had cast off earlier. The senate then voted him the same honours and offices - the tribunician power and governorship (and imperium). So in theory, the senate designated the supreme power to Tiberius - but by this time it was largely a toothless beast that Augustus paid lip service and outward respect to. By the time Tiberius died and Gaius (Caligula) was proclaimed emperor - no one was alive who remembered the Republic. The offices became hereditary through to Claudius and Nero.

Source: Syme's The Roman Revolution.