We can answer this question in several different ways and we must keep in mind that there was not one single reason for the fall of the Republic. In view of this, let me give you three causes.
First, Rome was getting too large for a Republican form of government. This was why people were challenging the form of government in unprecedented ways. Think of the political maneuvers of Gaius and Tiberius Gracchi when they circumvented the senate altogether, of the conflicts between Clodius and Milo with their personal gangs. In short, the Roman constitution was being stretched and change was needed.
Second, leaders took matters into their own hands. They created personal armies and even fought each other. Of course, this lead to civil wars. For example, Marius and Sulla were enemies. Pompey and Caesar also fought each other. All of this death and destruction caused people to want change. It so happened that the change also was the eclipse of the Republic.
Third, we cannot underestimate the political skill of Octavian. He stepped into the chaos of the Republic and introduced a solution - namely himself. His political acumen ultimately led to the Empire.
The above answer is a very good one and addresses the major points in the fall of the Republic. However, I would add one basic underlying cause: Greed and its corrosive effect upon the old Roman virtue of res publica.
Why did Sulla march on Rome?
Because he wanted the lucrative Eastern Command for the potential war with Parthia that had been stripped from him through political maneuvering and given instead to Marius.
Sulla wanted the riches, and more importantly, the glory a war with Parthia would bring and he would brook no opposition in regaining what he felt was his, and this included breaking Rome’s oldest and most sacred law, by entering the city of Rome with his Army.
Why were the Gracchi assassinated?
Because they had proposed returning the Public lands to the State and distributing them among the unemployed in Rome.
These lands had been taken over by rich Senators and formed into vast farms worked not by free Romans but slaves. The dispossessed flocked to Rome looking for work, and finding none, contributed to a chronic unemployment problem.
Indeed, Julius Caesar was assassinated, not because he was a tyrant, but because of his land reforms. Caesar proposed paying the Senators fair market value for the Public Lands, land that was not theirs in the first place, which would then be distributed to the unemployed plebes and discharged veterans. Additionally he required that the vast estates employ more Free Romans than slaves in their work force. However, the Senators were not having any of this and killed Caesar under the cloak of “restoring liberty to the people and saving the Republic.”
Why was the Triumvirate renewed in 55BCE (much to the consternation of the Senate)?
Because the three members, Caesar, Crassus and Pompey each received lucrative provincial commands that promised not only money but also glory. It should be remembered that to the Romans, glory and respect were just as valuable as gold and land.
Crassus was already the richest man in Rome, and Pompey a celebrated general, but both were falling behind Caesar who was accumulating both treasure and glory in Gaul. Their new deal gave Crassus the Eastern Command and a war with Parthia while Pompey received Spain with its vast silver mines, and Caesar - five more years in Gaul thus allowing him to consolidate his conquests there and avoid prosecution in the courts by his enemies in the Senate. To gain these laurels the Triumvirate was willing to use political violence as practiced by Clodius, Milo and their armed gangs.
This greed, pervasive and corrosive, was not new. Romans had always been competitive in all aspects of life, since the founding. However this had always been held in check by the Roman virtue of res publica.
Literally ‘the public thing’ it was a very simple concept: A Roman will do what ever is necessary for the good of the Republic, in peace and in war, in his public service and his private life.
No Roman better personified this virtue than Publius Scipio Africanus, and it was the genius of Octavian (later Augustus Caesar) to restore res publica as the guiding principle of his Principate.
The Assassination Of Julius Caesar – A People’s History of The Roman Republic by Michael Parenti, c2003
Caesar – Life Of A Colossus by Adrian Goldsworthy, c2006
Augustus by Anthony Everitt, c2006