It's actually somewhat difficult to discuss the early history of the Roman Republic, given how much of its history is mixed up with myth. As the Romans understood it, the Republic was preceded by rule of kings (beginning with Romulus, the founder of Rome, and ending with Tarquin). The last king was expelled and the Republic founded in its place.
Ultimately, the Republic evolved over the course of centuries. Roman history was largely shaped by the social conflict between the Patricians (Rome's dominant political and economic elite) and the far more numerous Plebians, with the Patricians having gradually been forced to grant concessions to the Plebians across the course of Roman history. The publishing of the Twelve Tables and the creation of the office of Tribune of the Plebs can be counted as critical moments in this history of social conflict and political evolution.
With the creation of Empire, these older political, economic, and social structures were transformed. Particularly important in this history, and in shaping the long term dissolution of the Republic, were the Marian Reforms, which involved a transformation of the Roman army, a service in which had been traditionally tied to property. Thus:
The army ... consisted of annual levies of amateur, peasant soldiers who were expected to supply their own weapons, food, and clothing, commanded by amateur officers who served a year and then returned to private life. ... Service in the army was seen as a duty and privilege of full citizens only, and those lacking the requisite amount of property, which constituted their "stock" in the corporate state, were excluded from this privilege (D. Brendan Nagle, The Ancient World: A Social and Cultural History, Prentice Hall: Upper Saddle River, New Jersey: 2002, p. 315)
However, to meet the demands of empire, Marius opened the military up to citizens without property. These reforms had the effect of tying the loyalty of Rome's soldiers more closely to their commanders, a fact which is powerfully illustrated in the history of the civil wars, and Caesar's own rise to power.
That being said, do note that while Julius Caesar plays an important role in this history, he did not himself mark the transition to Empire. After his defeat of Pompey, he did gain political supremacy in Rome, but he was killed in 44 BCE. Afterwards, there would be another round of civil wars, resulting in the rise of Octavius, who would become the first emperor, Augustus.