Broadly speaking, the status of the Roman emperor can be broken down into two periods: the Principate and the Dominate.

The Principate was inaugurated by the first Roman emperor, Augustus when he insisted on the title of princeps (which translates to "first citizen") and although he had de facto control over the entire empire, de jure he was simply first among equals. Julius Caesar was assassinated, in part, because his perpetual dictatorship blatantly indicated to the senate that Caesar intended to rule unilaterally mostly through the strength of his army--a mistake that Augustus planned on avoiding. Instead, Augustus decided to derive his legitimacy through the senate, and through carefully manipulating existing constitutional titles and roles, managed to avoid overt monarchical comparisions. The republic was dead, but the Senate did still have some power and relevancy, so it didn't die overnight. Augustus reigned for 40 years, and by the end of his rule the facade of republicanism was still up, but it was pretty well acknowledged where the real power was (you can see this in a lot of de facto modern autocracies as well, where the government is ostensibly democratic, but it's clear that power is clearly oncentrated in the hands of one person).

As time went on, the power of the senate wained and it eventually became appararent that emperors could no longer derive legitimacy through the senate's proclamations. Indeed, throughout the crisis of the third century, victorious generals were hailed as emperor by their troops without any regard of whom the senate considered to be the reigning emperor. This legitimacy through force understandably left the empire very unstable as any general with enough troops could claim to be emperor.

Eventually Diocletian rose to power and decided to resolve this problem by deriving legitimacy through divine right rather than an irrelevant body of old men (it almost goes without saying that Diocletian didn't have much love for the senate of Rome). By completely sidestepping the senate, Diocletian made it clear that he was not simply first among equals, but rather a divinely appointed lord, and thus ushered in the era of the Dominate. Whereas in the Principate, emperors were expected to be relatively available and modest, the Dominate was characterized by majesty and opulence. Diocletian had to create the sense that he (and his other co-emperors) were truely above everyone else, so that everyone else theoretically wouldn't be able to claim to be a legitimate emperor. At this point, Rome was clearly no longer a republic.

Keep in mind, the old Republican institutions and roles didn't really go away through this time period, but as time went on they became more and more irrelevant and more confined to the actual city of Rome. Many of these institutions even survived the fall of the Western Roman Empire, for example Consuls were still regularly elected even into the 6th century.

Sources:
Mike Duncan, The History of Rome Podcast (2010)
Karl Loewenstein, The Governance of Rome (1973)

Just to flesh out some of cjt's previous answer, I'll point out some of the ways pride in the republican form was reconciled with de facto imperial governance:

The Roman emperors had claimed some sort of divine right well before the Dominate, but it was masked in various ways. Augustus classed himself as the heir of the Divine Julius Caesar. A cult grew around Augustus himself as a living deity but was declared 'acceptable for non-Roman citizens only' (many parts of the Empire, particularly the eastern nations, had a long tradition of treating Roman governors as some sort of divine presence even in the Republican era). However the apotheosis (ascension to being a god) of each Roman emperor upon their death became a firm tradition, and the position of the highest priest in Rome (Pontifex Maximum) became part of the imperial office.

The extent to which that divine right was accepted varied between social classes and geographical regions. Emperors that were disliked had their divine pretensions mocked by the Roman social elite, often with an associated pride of a republic in which the Emperor was nothing but first citizen - for example Suetonius indicates that even Augustus was criticised for hosting a dinner party dressed as Apollo, and Caligula for similar actions. However this was generally specific criticism of an emperor, not of the tradition - e.g. Seneca's parody of Claudius' apotheosis includes a scene of other gods, including Augustus, criticising him. So the language of republican tradition was invoked when it was useful or under a particularly poor emperor but not under others.

On the self-reflective point, various writers reflect different views:

• Velleius writes in the early years of the Principate under Tiberius, and calls Augustus a 'republican princeps'.

• Tacitus, writing about 109 AD, bitterly criticised the perversion of republican traditions to the sycophany of a monarchy.

• Suetonius, writing about AD 121, under Hadrian, appears to regard it as a kind of dynastic monarchy, starting from the time of Julius Caesar.

• Cassius Dio, writing about AD 200 - 222, openly calls it a monarchy established by Augustus and approves of the change, saying that the Romans could no longer survive under a republic.

(Diocletian came to power in AD 284)

So broadly I'd note that, yes, Roman society did recognise that it had changed fairly substantially with Augustus, including during the Principate. Republican language was used as a point of criticism against unpopular emperors, and much of its structures and offices continued to exist but there were few illusions about where power lay.

Sources:

• Religions of Rome by North, Beard and Price
• Between Republic and Empire, ed by Raaflaub and Toher