The title above is fairly broad, so more specifically, what specific events led to the Warring States period experienced towards the end of the Western Zhou Dynasty in ancient China? As the lineage and vassal systems are broad answers to this question, what specifically about them contributed to this period in the otherwise illustrious and steadfast Zhou Dynasty?
The downfall of Zhou began with the end of Western Zhou, over 500 years before Zhou's final demise. King You's actions in seeking to please his concubine Baosi (including replacing the heir apparent with her son) led to a revolt by the forces of Marquis Shen and assisted by the Western Rong and Quan (Dog) Rong tribes. King You was killed and the capital Haojing was razed. The Eastern Zhou, established in Luoyang to avoid the Rong threat, never had the power to control its vassals, who took to fighting amongst themselves. The multitudes of states in the Spring and Autumn period consolidated into seven major powers in the Warring States and ultimately left Qin as the victor. Zhou's ultimate demise was due to a revolt against Qin, when, in alliance with other states, it launched a surprise attack on Qin. They responded by attacking Zhou, whose ruler quickly submitted.
Qin as a feudal power began when Duke Xiang received a fief and title from the new Eastern Zhou ruler King Ping. Three generations of his family had been engaged in the fight against the Rong and Qin were rewarded for their success (Sima Qian also mentions Qin's attempted rescue of Western Zhou and assistance during the movement to the new capital, which, while not quoted amongst King Ping's reasons for the reward, would likely have improved King Ping's opinion of them). They strengthened their base by conquering further barbarian territory and swallowing up some small neighbouring states.
The other major states of the Warring States period were all founded early in the Zhou dynasty, which presumably gave them the advantage of having had a long time to build up their power. I lack the sources to be able to properly explain how they were able to build a base that exceeded in power both the Zhou and neighbouring states that may have been of equally long standing.
@lins314159 is correct: the Eastern Zhou's fall began with the demise of the Western Zhou. Unlike its predecessor, the Eastern Zhou was never able to control its vassals. The precipitous drop in royal authority was caused by the collapse of the king's moral authority, his ineptitude, and the internal bickering of the court. This doomed the Eastern Zhou by creating a power vacuum for ambitious vassals to expand their own territories at the expense of royal power, kickstarting a vicious cycle of decline.
Following the eastward move, Zhou's moral authority and general prestige crumbled. But this has less to do with the barbarian invasion itself than one might reasonably think1. As @lins314159 mentioned, King You was killed by a coalition of rebel lords and barbarians led by the Marquis of Shen, father-in-law to King You and grandfather of King Ping.
However, the future King Ping fled to Shen in 777 B.C. as the Crown Prince of Zhou, three years before being disinherited and five years before fighting erupted. Thus, when King You was killed in the ensuing conflict, it exposed King Ping to charges of patricide and regicide - the two most serious crimes of Ancient China.
The eastern states regarded him as a usurper, an illegitimate king who came to power through murder. King You's (factual or otherwise) failings did not excuse this treason. The basis of Zhou governance had been a strict adherence to hierarchical rules, but in a single stroke the new King had broken two of its most sacred laws.
Henceforth King Ping, and the line of Eastern Zhou kings he sired, could command scarcely more than vestigial respect.
Internal Zhou Division
As mentioned earlier, King You disinherited his son before he was killed. This meant that King Ping was not the legal heir when he was being crowned by his grandfather.
The kingdom had already become divided between father and son, and the destruction caused by King Ping's regicidal rebels and the barbarians he (or his grandfather) invited failed to win him popular support. Instead, Zhou was plunged into a succession crisis as the nobles crowned the late king's brother2 as King Xie.
In addition to hemorrhaging the devastated Zhou demesne's resources, the conflict also ensured King Ping was wholly dependent on the support of powerful lords - who, as Ch'ien Mu pointed out, each had their own designs. At a time when the royal court needed to demonstrate strength and leadership more than ever4, the Zhou king was instead preoccupied with bestowing gifts, powers and honours upon its nominal vassals. Due to the unfortunate timing, this produced long term strategic implications. Zhou's patron states, such as Jin and Zheng, was able to leverage their political advantage into concrete territorial gains.
Duke Wu of Zheng, for example, egregiously breached the Zhou constitution5 by attacking and conquering two of his fellow vassals. Fully exploiting his position as a minister of the court, Zheng even commanded royal troops into battle against his personal enemies, and to annex land to himself. In this way Duke Wu forged the newly founded state of Zheng6 into the hegemon of early Spring and Autumn - in stark contrast to the precipitous decline of Eastern Zhou under his stewardship.
This usurpation of royal authority undermined Zhou's heavily impaired prestige, and contributed strongly towards the increasing expansion of the eastern lords. At the same time, the paralysis of royal leadership at a time of pressing barbarian threats, also provided a vacuum for the more powerful vassal states to step in and take up the mantle of defending Hwa-Hsia.
King Ping's Longevity
Zheng's meteoric rise to power demonstrates what a more capable leader, had one been in charge of Zhou, could have achieved. Unfortunately, not only was King Ping of ill repute, he was also incapable7. Worst still, he was also extremely long lived: King Ping's 50 year reign of stagnation fatally undermined Eastern Zhou's chances.
When he finally died, his grandson succeeded him and sought to rebuild Zhou authority by rebuking Zheng. Soon open warfare erupted, and although Zhou was still powerful enough to muster an army, its disastrous defeat in 707 B.C. ended all hopes.
 On paper, the Eastern Zhou (in its early years, at least) was not materially much weaker than its predecessor. Zhou's eastern demesne, around the new capital, was sizable and wealthy. Much of its western demesne, around the old capital, was reconquered by Qin's armies and restored to the royal court within two decades before being squandered as rewards for Jin .
 Previously thought to be another son of King You.
 There is debate over whether this meant the ninth year of King You's reign, the ninth year after King You's reign, or King Ping's coronation did not take place until nine years after the Duke of Jin killed King Xie.
 During the Spring and Autumn Era, conflict between the Chinese states and the barbarians kingdoms/tribes intensified. The organised State of Chu arose in the Yangtze basin in the south, while smaller, roving tribes of Di threaten the Central Plain states from the north. According to the Commentaries of Kung Yang:
 The powers of war and peace were supposed to be reserved to the King, who would issue orders for a vassal to attack. The king could delegate the power for a region to a trusted vassal, but this was normally done for remote locations because communications were difficult. The state of Zheng was right next doors to the Eastern Zhou.
 Zheng was only created in 806 B.C., and like Zhou itself migrated into the region from the West. After King Ping moved the court east, Zheng conquered the Eastern Guo and Kuai within four years, establishing its new territories in their former lands.
 King Ping eventually tried to curb the overstepping Duke of Zheng's powers at court. This was discovered by the Duke, and upon being confronted, the king gave up his own son to Zheng as a hostage for good behaviour. This was viewed as the king lowering himself to the same level as his vassals.
It all came down to greed and power. The Zhou Dynasty had become so large that it was becoming difficult, if not impossible, to manage and keep under control. As a result, the Zhou leaders created separate territories and appointed leaders over each territory. As these territories began to grow and prosper, they started to pull away from the Zhou leadership and started to form their own political structures.
Eventually, one of these territories (Ch'in) emerged as a true power with their own government and military. They eventually conquered each of the other territories of the Zhou Dynasty, effectively bringing it to an end.
The Zhou dynasty was brought down by the rise of the Qin dynasty, leaders of a ruthless but efficient state based in the west, that offered the "latest" in commercial, agricultural, and military technology.
The Qin soon imploded because of the unpopularity of its cruel regime, paving the way for the Han dynasty. But it destroyed a lot of what had come before it.
All these answers are to focused on politics. Agricultural technology advanced a lot under the Eastern Zhou dynasty. Better farming output means more people can live on the same farm, so population grew greatly.
The traditional, hierachial Zhou society couldn't handle this. The nobles, who were from the Zhou tribe, were forced to open their ranks to the commoners, who were conquered subjects. They weren't that respectful of their masters, so the whole empire started weakening.
The population pressure also made the states to start waging war against each other for land as a result. Since Zhou was losing control of China, it couldn't pita stop to the expansions. Then, it was only a matter of time before one of them became too powerful to be controlled by the emperor.