It is not possible to speak of only two events that led to the fall of the Roman Republic. There were many reason that lead to the downfall of the Roman Republic. In light of this, let me give you what I believe is the two greatest reasons for the end of the Republic.
First, the best answer is probably found in the fact that there were many civil wars during the end of the Republic. There was incessant warfare among the Roman elite. For example, Marius and Sulla were at odds and this was a precursor to what we would see between Pompey and Caesar. The destruction of civil war made it ripe for one person to take over.
Second, the political system or the Roman constitution became too complex for such a massive empire. There needed to be more centralization. The emergence of a emperor filled that vacuum.