Julius Caesar contributed to the fall of the Roman Republic in many ways, but most seriously in three. The first was by beginning the civil war from which the Republic never really recovered in January of 49 BC. Caesar had been proconsul (governor) of Gaul for ten years and had used this position to launch massive conquests, attaining wealth and prestige and making him as powerful as anyone in Rome. His opponents in the senate wanted to prosecute him; however, Roman law did not allow anyone holding a magistracy to be prosecuted. So Caesar was attempting to hold onto his command until he could be elected consul; his enemies didn't want this, not only so they could prosecute him but so he wouldn't have the ability to get another military command. After a lot of complex negotiations that eventually broke down, the senate declared an emergency decree against Caesar and his partisan tribunes fled town. At this point, Caesar began the civil war that would range across the Mediterranean for the next four years, from Spain to North Africa to Greece to Egypt. Although Caesar would essentially finish fighting these in 45, these civil wars never truly ended until 30, when Rome's first "emperor" (though he would never call himself that!) Augustus beat his enemies and began to establish a new political system, which we call the principate/empire.

The second reason Caesar can be said to be responsible for the fall of the Republic is what he did when he won. Throughout the civil wars, Caesar used the temporary position of dictator. This was a traditional magistracy (though one that had fallen heavily out of use, with the exception of Sulla's abnormal dictatorship after his civil wars in the 80s), meant to be used for a maximum of sixth months to resolve emergencies. By the end of 45, Caesar was declared dictator for life, making him the most autocratic figure in Roman history to date. He also did other unprecedented things that challenged traditional Republican political culture, like putting his own face on coins while still alive.

And finally, Caesar was responsible for bringing to prominence many of the men who would be important in the next set of civil wars. Most notably, two of his generals, Antony and Lepidus, were members of the second triumvirate. Some of his assassins even rose to prominence serving under him, like Decimus Brutus (not to be confused with the more famous Brutus). And of course, Octavian (later to be known as Augustus) could begin his rise to power because Caesar adopted him posthumously in his will: Gaius Octavius became Gaius Julius Caesar Octavianus, and his position as Caesar's son was incredibly important for him in the coming civil wars.

For further reading on Caesar, Adrien Goldsworthy has a great biography of him geared towards general audiences. He also has a recent biography of Augustus you may find interesting. For more on the horrors of the civil war period, Osgood's Caesar's Legacy is a fascinating (though more scholarly) book. If you want to look at the primary sources (which my answer was based on), the most important ones are: Cicero's Letters, Caesar's own account of the civil wars, Appian's Civil Wars, Plutarch's biographies of Caesar, Pompey, and Antony, Suetonius' biographies of Caesar and Augustus, and Cassius Dio's Roman History.

Edited to add: As always in scholarship, there's a lot of debate around this question. Not only about whether Caesar or his enemies were more responsible for the civil war, but about how inevitable the fall/transformation of the Republic was by the time Caesar marched on Rome. Was he just a catalyst, or was this all his fault? Morstein-Marx and Rosenstein have a great chapter on the various theories for the fall of the Republic in the Blackwell Companion to the Roman Republic.

TLDR: Caesar started civil wars that lasted for nearly eighteen years, made himself dictator for life, and was responsible for beginning the careers of the next generation of Romans who would fight wars against each other to attain supremacy and autocracy, including Octavian/Augustus, the founder of the principate/system of "emperors."

Well he marched on Rome and declared himself dictator for life, that's nothing new in Roman history though. Marius and Sulla marched on Rome a few decades before Julius did. To save you from a long history lecture, Marius marched on Rome, and Sulla marched on Rome to stop Marius. Marius and Sulla, mainly Sulla, brought the idea of sole power in Rome to ambitious people like Caesar, Octavius, and Antony. But in my own personal opinion, I think that the things that makes Caesar different from Sulla is that 1. He didn't relinquish power like the Sulla. Sulla wanted to fix the republic and leave the office after a few years, Julius Caesar wanted it for 10 years, then life. 2. There was someone behind Caesar to take over for him after his death. Augustus, as the first emperor and founder of the Julio-Claudian dynasty was the person to succeed Caesar and carry on the dynasty for the inherited monarchy to continue.

TLDR: Julius Caesar stayed in power, and somebody came after him.

I hope this answers your questions and if you need anymore info just ask. I'd recommend The 12 Caesars by Micheal Grant if you want a bit of an overview of what Julius and the other emperors did. Also if anyone wants to correct me on this, feel free to.