The main thing that Octavian (later to be called Augustus) did to change the political institutions of Rome was to take essentially all of the political power for himself. At the same time, having learned from what happened to his adopted father, Julius Caesar, he maintained the appearance that the Senate was an important body.
At first, Octavian called himself princeps, the first among equals and made it seem like he was co-ruler with the Senate. He kept the Senate as the maker of laws, but tended to screen the laws before they were passed by the Senate to make sure they were acceptable.
After that, Octavian used a number of devices to give himself legal cover for having so much power. He had himself named as consul. After that, he had himself named tribune of the people. Either way, he kept power for himself.
Octavian changed political institutions slowly and subtly. He did not call himself dictator or take any other showy titles. Instead, he took old titles and kept the Senate, but still controlled the process as de facto ruler of Rome.