What is the history of the Roman Republic as a political system from its foundation to its overthrow by Augustus?

With the overthrow of Lucius Tarquinius Superbus, the Roman Kingdom ended and the Roman Republic began. The rebellion against Superbus was undertaken by his privy council—known as the Senate—composed of the heads of families ennobled by the crown (known as patricians).
The Senate decided to abolish the monarchy altogether and, instead, vested executive authority into two co-equal consuls who would be elected by it for one-year terms. Within a few decades, the Senate itself was challenged in a plebian uprising and was forced to devolve some of its legislative authority to citizen assemblies whose support would be required to make law.
The consuls were limited in their power due to the dual nature of the executive, which required their mutual agreement to exercise the power of the state. (In the ancient world, other dual executives included the two co-equal kings of Sparta.) However, in times of war or emergency, full control could be vested in a dictator, temporarily elected for the duration of the exigency.
The Republic began to break apart following the consulship of Caesar. At the end of his term, Caesar was made a provincial governor and—upon his return to Rome—elevated to dictator for an unprecedented ten-year term. His assassination was followed by a seizure of power by Augustus and the start of the imperial period in Roman history.
https://chicagounbound.uchicago.edu/cgi/viewcontent.cgi?article=1300&context=public_law_and_legal_theory

https://www.ancient.eu/Roman_Government/

https://www.britannica.com/place/Roman-Republic