I would see a pretty definitive fall of the Republic, ironically, in the death of Caesar. Not because Caesar was some hero of democracy, mind you, but because his dictatorship was ultimately still predicated on the continued functioning of the entire Republican process - including the election of many of his enemies to office without interference from him. You can argue, of course, and not without merit, that Caesar felt firmly positioned enough to allow this rather than it being out of any concern for the greater stability of the polity, but the point is that Caesar's dictatorship, like the strongman regimes of Marius, Sulla, and Pompey before him, still maintained essential Republican structures.

With the Second Triumvirate, much of that was washed away, especially as Augustus accumulated power. If Caesar's dictatorship was an autocracy that violated the law while preserving the power structures of the Republic, then Augustus's autocracy violated the power structures of the Republic while preserving the law - and the law means nothing without power structures willing and able to enforce it.

However, Caesar's own dictatorship and the succeeding Second Triumvirate is, itself, only the culmination of almost a 100 years of political chaos, dating back to the conservative assassinations of the Gracchi. So I would definitely agree that it was a long process rather than any single point.