To start off understanding the shift from ancient Roman Kingdom to Republic, I suggest that we bear in mind an initial structure, based on a class system of King : Aristocracy : Plebs : Slaves. To this we have a strata of “clients” which we can more or less ignore from the point of view of government of the city.
It is quite probable that the introduction of a republican system took some transition so that the first post-kingdom leaders, like Lucius Brutus who ousted the last king Tarquin the Proud, were de facto like kings although clearly they had to watch their backs from being accused of it.
With the elimination of the King, the obvious thing was for the aristocracy, the Patricians, to use their wealth and power to fill the vacuum. They did, and through control of the Senate (the council of elders/aristocracy) they made laws which tended to defend their power. As one-sided as this was the Plebeians eventually managed to shift the system towards something a little more egalitarian.
A number of institutions developed in order to give Patricians, Plebeians and the Military a voice in government. The government itself developed a bureaucracy and positions of varying power which were open to this or that social class.
Persons campaigning for political position would spend personal fortunes in order to win votes. Julius Caesar himself almost ruined himself when as Aedile of the city he organised great circus displays for the joy of the people. By the time he was moving on from that position to govern Spain he had many creditors on his heels and was only saved by his close friend Crassus – the richest man in Rome.