Technically, the republic did not end. Octavian (and his "heirs") was said to have "saved the republic" and ruled by "will of the Senate" until a future date, when he would return his extraordinary powers. All of Octavian's successors carefully maintained this lie (at least at the beginning). The Republic began its death in 133 BC with the killing of the Gracchi brothers. Their deaths signaled the end of debate and legal procedure.
From that point on, whoever was willing to go the farthest dictated policy. Murder became commonplace during election time, and mobs were often whipped up by opposing parties to frighten enemies into submission. It became accepted, even encouraged, to use force to 'preserve the Republic'. The legions of Rome physically dismantled the Republic, but it was the Senate that set up a world where such a thing could happen as the citizens looked on and cheered.
Senators who could not legally block reform used assassination and trumped-up criminal charges to stop it; reformers who could not legally pass their bills used the steadily growing anger of the Roman populace to terrify the Senate or appealed to powerful generals and their armies for military support. Each time someone used violence to achieve an end, someone else hit back even harder to counter it. When Marius used his army of gladiators, slaves, and plebeians to seize Rome, Sulla hit back using professional legions. The result was a short-term stability and further weakening of the underlying structure of government.
The change also became one that put the men before the Republic -- no longer was it possible to survive in the new vicious world of Roman politics by being humble and loyal to the ideals of the ancestors. Powerful politicians vied to become "Primus inter pares" - "First amongst equals" through whatever means necessary, and ambitious men were only kept in check by other equally ambitious competitors. Marius and Sulla were the first, and their example gave rise to the first Triumvirate of Caesar, Crassus, and Pompey, and of the second one composed by Octavian, Antony, and Lepidus.
The Senate had proven, time and time again, to be so selfish, arrogant, incompetent and shortsighted that the Roman population no longer trusted them to lead. The Senate was often too willing to protect its friends, allies and members from lawful prosecution for even the most evident and extraordinary crimes; and because of this it lost the trust of the Roman citizens at large. When someone did come from their ranks and proved himself capable, the Romans flocked to them in a desperate hope that he might pull together the Republic and restore peace, law, and order. The Senate, using what means necessary, struck down these champions one by one, starting with the Gracchi. Each time this happened, the Roman people became more willing to accept the extreme measures of the reformers to ensure their laws, and their lives. Caesar's crossing of the Rubicon was technically treason, but no one outside the Senate cared, because it promised real change for a corrupt and unworkable Republic.
Within the Senate itself, the heavily entrenched, tradition-bound, rich conservative party was constantly at odds with any reformer that arose. The Gracchi worked outside the constitutional system by using the popular assemblies instead of the Senate; Marius had to fight tooth and nail just to get the necessary changes needed to recruit lower class soldiers; Sulla terrified the Senators with executions to enact reforms that were intended to actually preserve the powers of the Senate; and Caesar had to effectively conquer the whole Roman dominion in order to pass laws that were at least a century overdue. The harder the Senate fought to keep the status quo, the farther the reformers were willing to go, until at last it ended in Caesar's dictatorship.
The distrust the Roman citizens felt for the Senate was evident in the reaction of the troops to their commanders asking them to commit treason. The legions were willing to follow their commanders because they had no special love for the Senate, who only refused them pay and often fought over their rights to receive land upon returning home from war. There was no time when a commander asked his men to march with him on Rome and they refused, not one time where legionaries sided with the Senate. They chose to rally around names like Sulla and Pompey and Caesar, not the antiquated ideals of a Republic that rarely worked for them. The only thing that kept them in check, was each other. The Senate's inability to see this new reality cost it dearly. The Senate could not and did not want to adapt itself to the changing power structure, and as a result was pushed aside by those who could.
The policy of granting citizenship to the Italian allies had slowed down, resulting in a social war (90-88 BCE) which resulted in citizenship for all those who had put down arms. After civil conflicts, which led to bloody battles, Rome was entire in Sulla's hands who revived the obsolete office of dictator that had not much in common with the old office which had maximum duration of 6 months. This office would be terminated only by his death. Although the Roman Republic would last another 50 years, Sulla pretty much sealed its demise. He stood as an example for future generals that it was possibl to tke Rome by force and rule it. During the 2 decades following his death in 78 bCE, 4 mean played the leading role of the final demise of the roman republic. Pompey, Crassus, Cicero and Caesar.
Pompey understood that talent and accomplishment are shadowed by image. Crasses used his tremendous wealth to gain power. Cicero was a great orator who had studied in Greece and Asia Minor. When a revolt was staged against him, he was given total power and illegally executed them. Caesar aligned himself with the masses and slowly was given complete power. He was a peaceful dictator, and his reign marked the end of the republic. He maintained the facade of a republic, though, even increasing the size of the senate and creating many colonies for this veteran soldiers and extending Roman citizenship into some provinces. He adopted the Egyptian calendar. His main charge, though, was in installing a military monarchy. He operated within the bound of the republic, though, and when optimates trying to contest him thy merely stripped themselves of their own power.
Inadequacies of the Republic
Rome's government was not designed to rule an empire. The Republic was meant to govern a city-state; one that was, even at its founding, growing in scope and power, but nevertheless only supposed to extend through the regions of central Italy. When territory was captured overseas, the Republic proved itself unable to effectively govern it. The provinces became fiefdoms of the governors, who proceeded to plunder them at will and engage in military adventures that did not have the approval of the Senate. These governors eventually took on Rome itself whenever they were threatened.
There was no system of accountability, no ancient tradition of dealing with corrupt governors -- the problem was new, and the Republic, so tradition-bound, would not change to handle it. Once the Republic became an Empire, only an Emperor could effectively rule it, not an oligarchic assembly. But it took nearly a century before that was fully realized. In the end, the failure to control the generals caused the downfall of Rome's Republic. When Caesar finally took Rome for himself, he was greeted with thunderous applause, because he, at long last, promised, and even delivered, reforms the Roman people had wanted since the Gracchi.