I get a little baffled at why people think the Republic was 'falling'. There had been instances of individuals seeking to rule over it for a long time before in one manner or another, so a transition to one-person rule might well be regarded as inevitable - but the essential truth is that the Republic was a more hazy concept than we generally assume anyway. Okay, with a nod to high minded principles, the Romans threw off monarchial tyranny and set up a form of Republic but don't let that word fool you. It was not a modern democracy at all.
Rome was administered by elected magistrates who were given bundles of power according to the title they held temporarily from an elite group. In other words, tyranny was not eradicated in favour of proletarian empowerment but instead managed among selected individuals. The entire raison d'etre, the civic duty of the elite taking care of the general public, was for some a guiding principle but for many something to pay lip service to in favour of self interest, so really the emergence of warlords and eventual takeover by one of them isn't that suprising after a period of inflated wealth from conquest.
Antony and Cleopatra were intent on dividing the Romano-Egyptian world between them and their children. They had already attended public ceremonies dressed as gods. Clearly Marc Antony was buying into monarchy in a big way, something that Octavian was able to use against him as propaganda.
Yet the important fact remains that even after the dust had settled the institutions of the Republic remained. The Romans continued to call their empire 'The Republic' pretty much toward the end, even though we have Marcus Aurelius describing himself as an absolute ruler and Diocletian declaring himself as one. Those who repeat the mantra of Augustus becoming Rome's first emperor fail to grasp that he reformed the Republic, not swept it away as a dictator. Yes, he reformed it in a way that gave him at least 50% of the political support, but one can rationalise that not just as a convenient 'ruse to power', but instead a very necessary policy of survival in a political bear pit.