~He didn't. Although some historians define the end of the Republic with the appointment of Julius Caesar as perpetual dictator and others use the appointment of Gaius Julius Caesar Augustus (nee Gaius Octavius Thurinus) as the first emperor of the Roman Empire to demarcate the end of the Republic, and yet others cite the Battle of Actium as the defining moment, all agree that the Republic had been in a state of decline and transition for generations. Once that Republic had expanded its territory to an unmanageable extent and once the "barbarians" (literally translated, "outsiders") had been "pacified", the armies had little to do other than fight each other. The military leaders wanted political power. The selection of the political leaders came to dominate the business and backroom inner workings of the Senate. The early hereditary aristocracy was being replaced by the emerging "plebeian aristocracy". Civil war had essentially become the normal state of affairs.
This was the situation extant when Caesar returned from his deliberately provoked Gallic Wars. He joined an unofficial and informal alliance with Marcus Licinius Crassus and Gnaeus Pompeius Magnus. Crassus died during the Parthian invasion and Pompey set about to neutralize (and remove) Caesar. Caesar, always craving more and more power, reacted predictably by refusing to surrender control of his legions to the civilian government (the military leadership had long since assumed de facto control of the Republic) and by crossing the Rubicon with them instead. Although the Pompeian alliance had more senatorial support and vastly more (albeit inferior) troops, and initially defeated Caesar, eventually Caesar drove Pompey into Egypt where he was murdered and thereafter Caesar defeated the Pompeian armies of Metellus Scipio in the Battle of Thapsus, and finally crushed the last vestiges of the Pompeian alliance at the Battle of Munda. Caesar was now the only force of note left and in a position to write all the rules, which he promptly did. Even so, the civil wars continued until Octavian took over and eventually defeated Marc Antony, the last Caesarian ally of significance. Having built his political power base on patronage and favors and having assumed ultimate control of all Roman legions, Augustus had himself declared "Imperator Caesar" and the Republican form of government effectively ceased to exist de jure, as well as de facto.
The internal unrest and civil wars had been ongoing for more than a century. It was inevitable that at some point, someone would happen along who had the political savvy and support and the loyalty of enough of the generals and troops to finally rise to be the first among equals. Caesar believed himself to be that man. In the end, Caesar lost, as had so many others before him, most notably Lucius Cornelius Sulla. Octavian also believed himself to be that man, and he built the military and political power base from which to proceed before totally committing himself to the effort. Octavian won. Someone had to in time and Octavian did. However, Octavian and Caesar were simply a continuation of a pattern that had been long extant well before either of them were born. The Republic had doomed itself into extinction over the course of decades and generations and Sulla, Caesar and Octavian were simply the actors on stage when the final curtain came down. If not them, someone else would have been there for the final act just as any number of their predecessors could have ended the play before the ultimate historical last act was performed. It was the policies, not the personages, the ended the Republic.
By the way, be careful of using the information you get on this site. Octavian was Caesar's adopted son. Gaius Octavius Thurinus was Caesar's great-nephew and chosen successor, but Caesar took no part in raising him. In fact, the adoption was posthumous and effectuated by means of Caesar's last will and testament. With the adoption complete in 44 BCE, Gaius Octavius Thurinus legally became Gaius Octavius Caesar. In 27 BCE the Senate awarded him the title "Augustus" (a new title created just for Octavian) and the honorific became part of his name. In the 3rd century CE, Diocletian amended the basis of the title "Augustus" and appended it to his own title and that of subsequent emperors as a legal imperial title to designate the power and authority of the office.
