I've been taught that the assassination of Tiberius Gracchus was a turning point in Roman political culture. I understand that his reforms were aiming to address the land grab by patricians who would buy up land that soldiers were unable to maintain and forced to sell. Besides my stated question, I'm also curious as to how exactly Roman patricians were able to acquire land, leaving veterans destitute or homeless, without significant discontent from the lower classes?
Finally, what impact did these two developments have on the Republic itself? How important were they to its eventual decline and the rise of the Empire?
Both are interlinked but first a quick correction, it wasn't just the patricians that were gobbling up land it was the senatorial class. At this time the difference between patrician and plebeian in terms of magistracies was gone, both could achieve the consulship. Better to speak of the nobiles, a term for those from families who'd achieved a consulship - the 1% of the senatorial 1%.
Why were the Roman smallholders (the bedrock of the Roman army) losing their farms? Because the Roman army was a militia designed for a city-state. This worked fine when Rome was just another such state jostling for dominance and the campaigning was in the vicinity or even within the peninsula. But Rome was now an empire and facing long-running guerrilla wars in Spain and Numidia. This meant a soldier could get called up and not see home again for years and years (you were called up for a campaign not for a year). With these very long and frequent call ups, farms either failed or men started to dodge the census. This meant as Rome was expanding its territory, it's recruitable pool of soldiers was steadily shrinking and forcing a greater reliance on allied troops (who were getting cranky).
The failure of the Gracchi meant no land reforms (and thus no reinvigoration of the declining pool of recruitable Roman soldiers) but also the legitimization of political violence. T. Gracchus was assassinated in a riot led by the pontifex maximus who was not punished for it. Same for his brother. This set the precedent of violence and thuggery as an avenue for getting your way.
When C. Marius needed to recruit an army for his Numidian campaign and couldn't gather enough soldiers he waived the property restrictions and accepted volunteers. For the urban poor who were by law excluding from service, this was a chance to acquire booty and more importantly get rewarded with land following a successful campaign. And owning land was respectability. The person who'd get them this land distribution was their general (since the senate showed no interest) and this resulted in the soldiers becoming clients of their general. The more clients a nobiles had the greater his auctoritas and under the Roman client-patron system, a patron gained status when he helped his clients. So it was in the general's political interest to see that his men got rewarded with land. The flipside is if he failed to get them their expected land grants then he lost respectability and faced political oblivion.
The last factor and really the fuse on the powderkeg of the sinking Roman Republic (if I can mix metaphors for a second) was Sulla. In 88 CE Sulla had been awarded the command for the province of Asia and was going to thump Mithridates for massacring tens of thousands of Romans. This was going to be a lucrative campaign, the east was filthy rich (compared to the west) and the stereotype was that they were a bunch of girly-men. This campaign would cover Sulla with glory and make him very wealthy. Then the aging Marius got a tribune to pass a law awarding the province of Asia to him (he was a private citizen at this time). Sulla, needless to say, did not take this well. He convinced his army that Marius would recruit a new army and Sulla's men would lose out on all that potential loot - and so they joined him (note that Sulla's officers didn't). Sulla marched on Rome with his army forcing Marius to flee. He then restored his command and murdered the tribune who passed the law and then headed east. Sulla proved that any general with an army at his back could make the rules. When he returned from the east with his second march on Rome, he defeated the Marian faction in open civil war and then proceeded to murder and loot his political opponents.
This was the culmination of all the above factors: legitimized political violence, client armies, and ambitious generals with a personal army. The Republic's fate was sealed, it was just a matter of time.