What was wrong with the Roman Republic that caused its demise?

I think /u/dumb_ misread your question to be about the Empire, there has to be existing answer from one of the flaired commenters. If you don't feel like searching, I'll give you a semi-informed response.

The Republic was a system that had been developed to govern a city-state but had been expanded to run an empire. Its problems had it's origins from its own success. Traditionally the Roman army had been recruited from a pool comprised of those that possessed a minimal property requirement - this meant the poor were largely excluded from service (aside from serving as skirmishers). Originally the legions were mustered for a summer campaign within Italy and be dismissed at campaign's end. The problem was that Rome expanded outside of Italy meaning the soldiers had to add a few months of travel to reach places in Greece, Spain, or North Africa. Furthermore they enlisted for the campaign and if the campaign took several years to wrap up then those soldiers wouldn't see home for some time. For the wealthy or well-off soldiers they'd have enough money and slaves to easily weather the absence. But the small-holder who made up the bulk of the recruiting pool, they began to suffer from the long (and frequent) absences and began to lose their farms - which meant they no longer met the property requirements and could not serve.

During the Middle Republic the Roman army were 50% citizen legions and 50% allied. By the Late Republic they were mostly composed of allied troops due to the lack of recruitable citizens. Those who still met the property requirements were getting tired at getting called up more and more. Meanwhile the wealthy were buying up those failing farms and building up their assets. So the rich had more land than ever and the newly landless poor flocked to Rome to seek out the dole or manual labour jobs.

So in 134 BCE Tiberius Gracchus allegedly rode through Etruria and saw massive estates all being worked by slaves. Among his goals was to reduce the landless population by assigning land from the ager publicus. This was land that had been seized from enemies and owned by the state and supposed to be assigned out - a law that had been ignored completely. Instead the ager publicus was being used by the elites providing them with additional cheap land. He also wanted to enforce another existing law that restricted the amount of land a person could own - also completely ignored but still on the books. So there was no "stealing" the land from the rich, Gracchus wanted to enforce existing laws. Now the land commission was to be staffed by him and his relatives - but this type of corruption was standard Roman politics and would boost his standing/popularity. Needless to say the senators who were using the public land weren't terribly happy about all this and tried to block his legislation. T. Gracchus submitted his law for approval to the senate but they refused approval - but this was not technically needed but had always been sought (the Romans liked their traditions) and T. Gracchus just decided to submit the law to the Plebeian Assembly (which would still be binding on the entire state). The senate got another tribune to veto the law and T. Gracchus riled up the crowd to toss the offending tribune out and had his law passed. The senate used the excuse that he was formenting unrest and planned to seize power as king and a few senators and their supporters attacked Gracchus and his supporters and murdered the tribune. No one was convicted of this and it established a precedent that you could use murder in politics and get away with it. His brother Caius Gracchus learned that negotiation was pointless (T. had gone out of his way to try to accommodate the rich) and pushed a hardline land reform program - even tougher than his brothers. He was killed by senators in a mini-civil war.

All that was to set the stage for the Late Republic. Now the entire republican system had never been intended to be democratic - the Romans deeply distrusted democracy as little better than mob rule. What they wanted was a balanced system but it was one where 2 of the three parties held the most power (consuls & senate). The comitia centuriata elected senior magistrates and declared war and peace and used a complicated system of voting. The populace was divided into 193 centuries organized by wealth - the poor would be grouped into the last few centuries. As each century got exactly one vote it meant a rich man's vote (in one of the leading centuries with very small numbers) was worth much more than a poor man's (who'd be crammed in the last century and his voting power diluted by the tens of thousands of other poor). In voting they went in order from wealthiest centuries to the poorest - and often the question (war or electing a consul) would have enough votes for assent that the poor never got an opportunity to vote. There was the plebeian assembly which elected plebeian aediles and tribunes of the plebs and they voted by tribes and in random order. This made it much more democratic and once the tribunes ignored senatorial approval they could push through legislation (usually land reform).

Topping this confused system off was the way Romans governed their provinces. For the elite, politics was everything and it was a zero-sum game. Losing an election meant you and your family lost honour and respect and so they went at it with gusto. Bribery became prevalent and holding games was a standard way of winning over public support. Both of these cost money and most Roman elites, while filthy rich, lacked liquid assets - they instead possessed hard assets (e.g. land). So to get elected they would borrow tremendous amounts of money to use for bribes and to hold games. Once elected they'd serve their year in office and then be dispatched to their designated province. Now none of these jobs (consul, praetor, etc...) paid a salary, you had to be wealthy to qualify to run and were expected to do this gratis to serve the state. So what inevitably happened was once out in the provinces, governors would do anything and everything to raise money to pay off their debts. The case of Verres is probably the most extreme example (even if he only did half of what Cicero accused him off) but probably representative of many governors who had no oversight and debts to pay. They'd cheat, extort, kidnap and ransom, or accept bribes - leading to a steady stream of complaints from the provinces and steady business at a court for political corruption. Allegedly Verres claimed he needed to make three fortunes while governor: one for him, one to bribe his judge at the inevitable trial, and one for his creditors. So those elite of the elite who were getting into the higher offices were spending ridiculous amounts of money to get into office and then raising ridiculous amounts of money (legally or illegally) which allowed them to continue the cycle.

So you had an ultra-competitive environment with millions of denarii floating around, constant wars (caused in part by ambitious generals looking for military glory to further their political hopes), a declining pool of recruitable citizens. Enter C. Marius, a 'new' man (no one in his family had been consul before - very rare in the Late Republic). He had been elected consul and assigned the war in North Africa but couldn't raise sufficient soldiers for an army. So instead he waived the property requirements and accepted volunteers - Rome was rich enough to supply arms and equipment now. A horde of the urban poor joined the standard because this offered an opportunity for pay and more importantly a chance to get land as a reward for their service upon a successful campaign. The person who could get them this land was their general - so now soldiers looked to their commander instead of the senate (who had always been reluctant to give up any of the ager publicus). This set the fuse in the powderkeg.

Marius was successful in his war - but it was an officer of his who engineered the capture of the enemy king. Now traditionally the general won the accolades for anything his lieutenants did. But Sulla wanted the glory of ending the war and Marius wasn't interested in sharing. The bad blood between them developed and over time would fester. Around 105, some migrating Germanic tribes entered northern Italy and wiped out a pair of Roman armies (led two feuding generals who didn't cooperate and were thus defeated in detail). There was mass panic in Rome and they elected Marius continuously as consul while he went up north and defeated the Germans. Now people had been elected consul more than once before, but this was a rare achievement - and now Marius had been elected 6 times. Another precedence had been set.

Marius however was soon pushed out of power and went away grumbling. We turn to Sulla, his former lieutenant. He rose up the political ladder and proved himself a good and ruthless leader. In 88 BCE Sulla had been given command of the war against Mithridates in Asia - this was a coveted command as Asia was filthy stinking rich and (to a Roman) no Asiatic could be a challenge for Rome. So this was an enemy on easy difficulty who guaranteed multiple epic drops. Marius saw this and wanted another kick at the post - despite not being a consul he traded favours with a tribune of the plebs to pass a law (through the plebeian assembly) giving him command of the Mithradaic War. When Sulla was informed of this change, he took it well. By which I mean he convinced his army that they'd be disbanded and lose out on all that potential loot as Marius would raise a new army (likely including his supporters) - and so he led his army to march on Rome. Marius did a runner (back to his power base in North Africa where his former veterans were settled) and Sulla had the law annulled (and the tribune who proposed it murdered) and then marched on to Asia.