I probably should not but, since we have no choice but to live through interesting times, I'll provide my perspective - a different one to the accepted answer.
Fair warning: keep in mind we're discussing history, not graphic design -- almost nothing aligns neatly in history. Roman history especially. I like it exactly because it's fuzzy and warm, enough to keep it friendly and me interested. My friends, however, have stopped asking for my opinion on the Romans because I don't have "answers", just ideas.
So, the beginning of the end of the Republic and how did Tiberius Gracchus contribute to it?
Answer: In simple terms, it wasn't the fact of the reforms (or legislation). It was how Tiberius Gracchus went about it.
Let me explain in three parts. First, Tiberius Gracchus used his powers as tribune to help plebeians (the lower classes, common folk). This was exactly what was expected of a "tribune of the plebs". He introduced reforms, principally a law to create more land for farming (confiscating land held by higher class and transferring them to the lower class), which would also solve other problems then facing the Republic. He was being a populist and it angered the higher class (taking their land especially).
But he did something else. He went against political convention (cf. concordia) which angered the powerful men of Rome (the senators) even more. When his initial ideas/proposals were rejected, he ignored the Senate (which, by convention, he should not have) and went straight to the Assembly for authority to enact his reforms (and also give himself power to, in effect, decide on whose lands he could confiscate). One could say he defied the senators. But it was more that that, Tiberius Gracchus undermined them and he undermined the accepted convention, or ancestral custom (mos maiorum), of tribune and senators working together amicably to solve Rome's endless headaches of governing a large republic.
Second, by undermining the Senators and the political convention of working together, Tiberius started another one - the convention of using violence to solve political disagreements. From SPQR: A History of Ancient Rome (2015) by Mary Beard, p. 216:
Looking back over the period, Roman historians regretted the gradual destruction of peaceful politics. Violence was increasingly taken for granted as a political tool. Traditional restraints and conventions broke down, one by one, until swords, clubs and rioting more or less replaced the ballot box. At the same time, to follow Sallust, a very few individuals of enormous power, wealth and military backing came to dominate the state – until Julius Caesar was officially made ‘dictator for life’ and then within weeks was assassinated in the name of liberty. When the story is stripped down to its barest and brutal essentials, it consists of a series of key moments and conflicts that led to the dissolution of the free state, a sequence of tipping points that marked the stages in the progressive degeneration of the political process, and a succession of atrocities that lingered in the Roman imagination for centuries.
The first was in 133 BCE, when Tiberius Sempronius Gracchus, a tribune of the people with radical plans to distribute land to the Roman poor, ...
He was killed at an election (to be Tribune for a second time), by a group of senators and their henchmen. Gaius Gracchus, his brother, carried on his reforms. Actually, he doubled-down, and initiated even more reform. However, unlike his brother but probably because of the violence against his late brother, Gaius Gracchus resorted to armed insurrection. It failed and he died in 121 CE. Then we get to the Social War, Sulla, the First Triumvirate and so forth - violence begetting more of the same (Martin Luther King, if I’m not mistaken, said something similar).
Finally, in 46 BCE, less than a 100 years after the death of Tiberius Gracchus, Julius Caesar was the last man, of the Triumvirate, still standing and with his horseguards (Germani corporis custodi) and Praetorians beside him, no one dared stop Julius Caesar from becoming a sole consul.
No one then saw it coming, at least not the ones who mattered. Or those who did foresee would have been killed (I assume).
This is a very short sketch of this political history. And it is never really one, or even a few key factors, to the rise or demise of any state. There will always be a multitude - of people, of events, and of opinions - to the course of history. And that is why this question is so appealing, at the same time so difficult to answer. Whilst we watch in real-time the gradual destruction of a republic, we can only speculate what caused the fall of the original republic.