There are many, many sources from the world after 476, the year when the Western Roman Empire allegedly fell. However, there is little that directly talk about people's daily lives, as what we have were overwhelmingly written by the elite of society, whose poems, letters and histories are far more concerned with matters of high politics than what the average man experienced. We do have some sermons and saints' lives that allow us to get an insight into people's daily lives, but they only provide limited snapshots and should not be seen as representative of everyone's experiences. As such, I would suggest that you start with a few modern historians' take on this period, as they synthesised some of the latest research in their fields to write readable narratives for the general audience, which should give you a far better picture about life after Rome than from just reading the primary sources. Chris Wickham's The Inheritance of Rome: A History of Europe from 400 to 1000 (2009) is a solid narrative history of this period, whilst Peter Heather's The Restoration of Rome: Barbarian Popes & Imperial Pretenders (2014) is a political history more centred on the idea of Rome and how imperial ideas continued to shape later events. On the other side of the coin, we have Bryan Ward-Perkins' The Fall of Rome: And the End of Civilization (2006), which argued that everything really was that awful in the fifth century. James O'Donnell's The Ruin of the Roman Empire: A New History (2008) was similarly melodramatic, but places the awfulness in the mid-sixth century instead. There is in fact a great deal of disagreement on what the post-Roman world was like, so keep this in mind when you read these books and you should think about their arguments critically. Do the same for my answer as well, as I lean heavily towards the view that there was a lot of continuity throughout this period.

So what do the sources actually say? An important source to analyse would be the Life of St Severinus, which you can read here. This is a hagiography of a holy man active in Noricum at the very end of the western empire and life was certainly very grim. An oft-cited passage is this:

So long as the Roman dominion lasted, soldiers were maintained in many towns at the public expense to guard the boundary wall. When this custom ceased, the squadrons of soldiers and the boundary wall were blotted out together. The troop at Batavis, however, held out. Some soldiers of this troop had gone to Italy to fetch the final pay to their comrades, and no one knew that the barbarians had slain them on the way. One day, as Saint Severinus was reading in his cell, he suddenly closed the book and began to sigh greatly and to weep. He ordered the bystanders to run out with haste to the river, which he declared was in that hour besprinkled with human blood; and straightway word was brought that the bodies of the soldiers mentioned above had been brought to land by the current of the river.

Such was the dishevelled state of the Roman garrison in the region that the locals eventually had to pay money to local barbarians to act as their protectors. This certainly sounds pretty bad, but we have to be careful when reading sources like this. This text was written to promote the cult of a holy man, not to depict reality accurately. I do not doubt that it reflected the honest sentiments of a refugee (the author wrote this in southern Italy), but I don't think it described the totality of everyone's experiences. For example, as far as I know there are no surviving sermons bewailing barbarian rule from this period, which compares favourably to the period I actually specialise in, the seventh century, when the clergy really went over the top about how the Arab conquests heralded the end of the world.

Sources like the Life of St Severinus presented a grim view of their times and as a result contributed to the popular view that the Dark Ages really were that dark, but if read between the lines, I think a different narrative emerges. At the end of the tale, King Odoacer of Italy attacked the Rugii, the ‘barbarian’ rulers of Noricum, but rather than merely conquering the province, he allegedly ordered the local Romans to evacuate themselves to Italy and bring their saint's body with them. This was done under the auspices of Count Pierius, a man with a very Roman title, and his ultimate superior, Odoacer, was not only the king of Italy but was also an important Roman general who now operated under the vague suzerainty of the Eastern Roman Empire. Once in Italy, these refugees were allotted different districts to live in, which is also suggestive of the considerable bureaucratic and material resources the kingdom possessed. This all sounds very Roman doesn't it? After all, the ‘barbarians’ had been decisively defeated and the Romans safely evacuated.

But how can this be if the Roman state had already collapsed by 476? As I alluded to in the first sentence, it is I think a mistake to say that 476 represented the fall of the Western Roman Empire. In fact, the first reference to that year being an important date comes from a Constantinopolitan chronicle from c.518, just as they began to consider attacking the Ostrogothic Kingdom of Italy, so it should be seen as a politically charged date that related strongly to Eastern Roman propaganda rather than a historically significant date a lot of people still think it is.^1 Instead, Odoacer's coup can be more accurately described as the revolt of a disgruntled Roman general - there was really no reason for him to dismantle the system entirely, which made him no different from everyone else who wanted regime change when the government upset them. As the New Cambridge Ancient History put it:

Within Italy, he did much to ensure the continuation of normal life, and those who enjoyed the fruits of his government were aristocratic and senatorial clans. Odoacer had won his position by force of arms, but he maintained a show of legality by seeking recognition from the eastern emperor.

This only escalated further when Theodoric the Ostrogoth defeated him in 489. Theodoric was the king of a ‘barbarian’ people, but he was also an ex-consul who had spent his formative years in Constantinople. His rule in Italy was therefore an exceptionally Romanised one. He was certainly well-liked and was lauded by Romans as a princeps, an augustus, and the "propagator of the Roman name", to the extent that perhaps his subjects even saw him as the new emperor of a renewed Western Roman Empire.^2 Even in a relatively hostile Greek source, Procopius' The Wars of Justinian, he was treated with a reasonable amount of respect:

Though he did not claim the right to assume either the garb or the name of an emperor of the Romans, and was called rex for the rest of his life (for the barbarians are accustomed to call their leaders that), still, in governing his own subjects he invested himself with all the qualities that are appropriate to one who is an emperor by nature. For he was extremely careful to observe justice, preserved the laws on a sound basis, protected the land securely from the neighbouring barbarians, and attained the highest possible degree of wisdom and courage. Neither did he himself commit scarcely any act of injustice against his subjects nor did he allow anyone else to attempt it, except, of course, that the Goths distributed among themselves the lands that Odoacer had given to his own rebels. In name Theoderic was a usurper, yet in fact he was truly an emperor no less than any who have distinguished themselves in this office from the beginning.

Of course, many of the sources praising Theodoric were written by people involved in his regime, most famously Cassiodorus, who served as the praetorian prefect (how Roman is that?) in the Ostrogothic court and who wrote a huge amount of official letters on its behalf, all of which trumpeted his regime's inherent Romanness. In this case, we also have to remember that these letters were later edited in order to suit the political climate in Constantinople,^3 but the Roman core of these texts, and indeed that of his compatriots, cannot be explained away simply through their pro-Theodoric bias. Cassiodorus’ contacts included senators in Rome, emperors in Constantinople, ‘barbarian’ kings in Gaul, Spain and more, all regions that continued to embrace Roman culture regardless of the events of 476.

This can perhaps be illustrated by looking at King Clovis of the Franks, the first Merovingian king to dominate Gaul, whose exploits included bashing a subordinate's head in with an axe in public, but who was nonetheless given the honour of being an honorary consul by the emperor Anastasius. Much of this was the result of the complicated nature of imperial politics, but there was certainly a hunger in the West for traditional Roman titles. Clovis' contemporary, King Sigismund of the Burgundians, was for example similarly rewarded with the title of magister militum (the master of soldiers) from Constantinople and did so with the help of a local bishop, who was seemingly all too happy to help his ruler achieve his goal.^4 Clovis was also a friend of bishops, including Remigius of Rheims, who had advised Clovis to protect Christians even when the king was still a pagan.^5 In most parts of the post-Roman world, ‘barbarians’ and Romans co-existed and found a way to live with each other, so we should not be surprised by this. The ‘barbarian’ kingdoms needed the taxation system to function, people to man their armies and even something as basic as people to socialise with, whereas the local Romans wanted security and prosperity, both of which the Western Roman Empire had failed to provide in its last decades.

End of Part I

Perhaps the most famous and comprehensive chronicle of life under Frankish rule is Gregory of Tours's History of the Franks, the landmark text of Merovingian Europe. Gregory (who lived from c. 538-594) was the bishop of Tours in present-day France. He was Gallo-Roman, which gives him a unique perspective on the Merovingian rulers of the time. He belonged to the senatorial-ecclesiastical class which still self-identified as Roman (there are a number of saints/bishops in his immediate family line) and so is often able to consider the Franks from the outside (drawing much attention to their notorious acts of violence/torture), while also (probably for political reasons) apologizing for much of this same violence.

The book (which is actually ten books) is a fascinating read, covering all of human history as it was then understood from creation through to the 590s, especially covering (of course) the appearance and rise of the Franks and their apparently endless internecine warfare. The text is actually very heavy on the 570s through 590s, since Gregory had experienced much of the history of that time period personally.