The answer to this depends to some extent on whether one believes in the "great man" school of historiography. I tend to believe that larger forces are what drive history much more than the actions of individual people. For this reason, I would argue that the death of Julius Caesar was simply the immediate cause of the death of the republic while the true causes were broader societal conflicts.
For more than 150 years, going back to before the reforms of the Gracchi, there had been serious divisions within Roman society. There was, for example, the divide within the upper classes between the opitmates and the populares. The first of these groups was made up of those elites who held the power in Rome. They wanted to keep power for themselves while the populares wanted to take power for themselves by using the pent up frustrations of the lower classes for their own purposes.
It was the continuing conflict between these groups, and the military conflicts that went with it, that truly caused the fall of the republic.