Answer:

The Roman Republic is probably best described as a pseudo-democracy of sorts. Its creation and initial set-up actually pre-dated Athenian democracy by a single year, though even until its dying days it was more of a "democracy for the privileged" than anything. Hence, Classical-period Athens is usually considered the world's first true democracy. Nonetheless, the Roman Republic was clearly a very successful model for a state. It was ultimately the lack of fail-safes and precautions against powerful consuls and leaders that led to its down-fall, and arguably the fact that only landowners had a say in the vote, or indeed only equestrians (members of the ancient noble families) could become senators or consuls. Your average peasant, born in Rome or otherwise had little say in the state.

Generally seen as the first real sign of the downfall of the Roman Republic was Lucius Cornelius Sulla's stint as dictator (to use the actual Latin word). Dictator was a special privilege granted to a military leader (general) by the Senate -- often a past consul by nature of the office -- in times of great need. For example, during the Sack of Rome in the 4th century B.C. by the Cisalpine Gauls (of northern Italy). It was however used extremely sparingly until Sulla was granted the position during the Roman Civil War of ~82 B.C. (the very worrying second one in the span of a decade). The senate relied on the honour of the general to relinquish the title of dictator when it was no longer required and return to normal life (of a senator often).

Fortunately at first, Sulla did this, though as we all Gaius Julius Caesar did not and craved power until his final assassination. Ironically, Caesar was known as a member of the populares (unlike his aristocracy-aligned rival Pompey [Gnaeus Pompeius] of the First Triumvirate), gaining widespread support from the working class, and thus facilitating the end of what small democracy there was, and the start of Octavian's tyrrany. I hope you can see from this that effecitvely the collapse of this pseudo-democracy was facilitated greatly by the lack of sufficient legal codes as well as increased internal strife within Italy, propagated by corrupted and varying degrees of oligarchic control. There were always too many loopholes in Roman politics and law to allow power-hungry men to wield disproportionate control. One might say the Roman Republic was a time bomb ready to explode (though not as disastrously as the Roman Empire of course!).

A lecture summary for a history course on Classical History in the Purdue University gives a good summary and highlights four main causes (in the lecturer's view, though largely accepted by historians I'd say):

The rise of the popular tribunes -- late 2nd century B.C.

The rise of the private armies -- most notably by Garius Marius (namesake of the notable Marian reforms of the military) and Lucius Sulla (the famous twice-dictator).

The First Triumvirate -- as mentioned above, consisting of Julius Ceasar, Pompeius Magnus, and Lucinius Crassus. Note that the second triumvirate effectively occurred after the fall of the Republic.

The granting of Caesar's dictatorship -- a 10 year period in which he was meant to restore the Republic, but instead laid the foundations for the Imperium beginning with Octavian (Augustus).

All contributed to increase oligarchy and eventual pseudo-monarchy in the Roman Empire. The natural response to the many crises that occurred over the first century B.C. was to put power in the hands of great individuals; a dangerous move at any time, despite it having some logic. No doubt, the spiral towards collapse had begun long before Caesar's heydey. Worth noting: the first Emperor Octavian cunningly maintained the pretence of the Republic's continuation, gradually dropping it towards the end of his reign.

Now there are of course other explanations, and I have only touched on the surface here, for it is a highly complex subject, but I hope this gives some insight for you to go and read around the matter further, at least.
References

Rubicon: The Last Years of the Roman Republic, Tom Holland (2003)

Fall of the Roman Republic, Purdue University (2011)