This is kind of a broad question as the time period when Rome was still under Constantinople's de jure control lasted for around 250 years. Furthermore the sources from Rome during this period are often scant and not very helpful, however I'll do my best to atleast cover the basics of Rome under Byzantine control.

Before I get into the meat of the answer, I feel a little backstory should be useful for those not aware of the political context in Rome during this period. When the Western Roman Empire "fell" in 476 to the Germanic general Odoacer, not much changed for Rome. Odoacer left the local administration intact and ruled as King of Italy from Ravenna just like the previous Emperors had. The Senate was allowed to retain local political control over Rome, elect new Consuls and continue uninterrupted with their politics.

When the Ostrogoths took over in 493, they very much continued Odoacer's system. However with the Ostrogothic rule we have much better and more thorough sources to work with, and most of the sources regarding Gothic rule are unanimously positive. Theoderic allowed the Romans and Goths to live by their own laws and local customs. Romans would be judged in Roman courts and Goths in Gothic courts.

The Senate gained an increased position under Theoderic, as the Ostrogothic King saw them as a useful tool to further his legitimacy as an Imperial patron, and not a foreign barbarian King. The Senate, so pleased with Theoderic even made him Consul and hosted animal hunts in the Colosseum to honor their new patron.

Many buildings were also restored during this period and new buildings were built. Local infrastructure was also repaired to sustain the local population and economy. The Goths were not seen as a foreign government, but as a legitimate imperial representative. Theoderic worked to ensure the Goths were not seen as foreigners, the Eastern Emperors referred to him as Patrician of Italy and bestowed many gifts and titles upon the Gothic King. As far as most Romans were concerned Italy, and Rome itself was still under Roman rule, it just had a Germanic Patrician the province.

All of this would change however in 535 A.D. The tolerant and open minded domestic policy of Theoderic had ended after his death, and as such many Romans wanted to return to Constantinople's fold. This war of reconquest however would not go as planned, and would ironically end up destroying the Roman culture and way of life in Italy. The war following 540 would become a conflict of attrition, plague, depopulation and brutal sacks. Rome itself was besieged 5 times in only 20 years and the city was completely depopulated twice.

Many Senators, accused of being spies or sympathetic to Constantinople were also taken hostage and/or killed. The sheer destruction of the war, and the hostility of the plague made many wealthy Senators flee to Constantinople, which was safer and more stable.

As the dust finally settled, the city was a wreck. Most of the population had left, the basic infrastructure was in disrepair and the plague was persistent. Justinian tried initially to remedy this by giving Rome a large amount of subsidies, to pay for it's maintenance and encourage the population to return. Looting was also an issue during this period, which is why Justinian introduced an edict to prevent this, said edict stipulated that anyone caught stripping public monuments would have their hands cut off as punishment. It's unclear if this edict was ever enforced.

The Senate still met during the 6th Century, but their wealth and prestige had been severely diminished. The days of Consuls celebrating their election with chariot games or animal hunts were over, there was no private wealth left to fund such extravagance in the West.

In 568 the Lombards invaded Italy, this invasion resulted in a massive weakening of Imperial power in Italy. Constantinople's de jure territories in Italy were reduced to a strip between Rome and Ravenna plus a series of coastal towns like Venice and Genoa. Because of this local power within Rome gradually fell within the hands of the Pope and the Imperial government became gradually less influential in Italy.

The future Pope Gregory travelled as an ambassador for Pope Pelagius II to Constantinople in 578 to ask for military aid against the Lombards. The Emperor, Maurice stated that his sympathies were with the Pope, but that the Empire could not send relief to Rome due to it's more pressing issues elsewhere. It thus became clear to Rome that it could no longer depend on Imperial aid to defend itself, Constantinople had more pressing issues in the east against the Persians and later Arabs.

It is therefore probably no coincidence that the Roman Senate dissapears from history during this time. The elites gradually moved to military positions in order to defend their territories. Constantinople could no longer help, so Rome had to take care of itself. The Roman Senate's last recorded actions was the erection of a column in the Roman Forum dedicated to the Emperor Phocas (602-610). The column, which was the last public monument to be erected in the Roman Forum still exists today. The column is in many ways a perfect microcosm of the times. The monument is generally agreed to be either a rededication of a older monument, or a column stolen from a earlier Roman building. The gilded statue on top was also probably spoila, of a previous Emperor. It shows that faith in the Imperial government was still high in Rome, but that this faith was in many ways largely imaginary, something held up because of tradition and formalities. Following the erection of this column in 608 the Senate dissapears, with the Senate House being converted into a church in 630. Their duties were no longer needed as the ecclesiastical and military authorities took over their former tasks.

Phocas was infact quite the influential Emperor in terms of Italian politics. He had accepted the Pope's supremacy over the Patriarch of Constantinople and established a temporary peace with the Lombards (as celebrated on his column). Phocas also donated the Pantheon to the Pope, who converted the abandoned temple into a church, probably saving it from destruction and allowing us to witness it intact today.

Rome was also victim to many natural disasters during this time period. There was a particularily devastating flood in 589 and plague hit once again in 590. Said plague gave birth to a legend which states that the Archangel Michael appeared on top of the Mausoleum of Hadrian and sheathed his sword to signal the end of the plague.

During the reign of Emperor Constans II (641-668), Rome took part in an extraordinary event. The city saw it's first Imperial visit in two Centuries. Constans came on friendly terms with Pope Vitalian and took residence in the Lateran Palace. However the Emperor, upon touring the city would proceed to strip all the public buildings and statues of bronze and other valuable materials, in order to aquire armament material against the Arabs.

However despite this, Rome continued to prefer Imperial rule, largely because much of Rome's food supply came from other parts of the Empire, like Sicily. The Pope was also given some flexibility and autonomy under Constantinople, which as said had minimal control over the region, the alternative, the Lombards would have much stronger authority if they took Rome and would be able to curb the Pope's local power.

However this relationship would fall apart during the Iconoclastic Controversy in the 8th Century. The Pope, furiously opposed to the destruction of icons excommunicated the Iconoclasts. Constantinople responded by attempting to capture the Pope, but this ultimately failed. From this point onwards Imperial control of Rome was barely even a formality and the relations between Constantinople and Rome were extremely hostile due to the disagreement over Iconoclasm. As such the Papacy would gradually break all formalities in regards to being a part of the Empire, and began to see the Franks as their new earthly protectors, ushering in the Holy Roman Empire.

So to summarize this post. Life in Rome during this time was characterized by a transition from secular, Imperial institutions to ecclesiastical institutions. The city was left on it's own without help from Constantinople and had to transition into a government that was better suited for self defense and de facto self rule. The city was heavily depopulated and most of the city had become abandoned.

The relations with Constantinople varied during this time period. During the 6th Century many citizens definetly saw themselves as part of the Empire and viewed the Emperor as their sovereign. However as Imperial power within Italy gradually weakened, people in Rome likely became more disenfranchised with Constantinople.

Disagreements on theology would serve as hotspots for political tensions and the Iconoclastic controversy was the straw that broke the camels back. Form this point on Rome and Constantinople were hostile and saw eachother as heretics. This would eventually lead the Papacy into the Frankish sphere of influence and the seperation from the Empire.

The Romans at the time definetly did see the so-called Byzantines as Romans, however following Iconoclasm, Rome sought to distance itself from Constantinople and as such began to deny their claims to the Imperial title.

I hope that answered the question. If you have any further questions or wish me to elaborate, just ask. It's difficult to summarize 200 years of history into a single post.