So the closest primary source we have to this is Suetonius's 12 Emperors (which we should still remember was writing after Caligula's death and thus is influenced by the same biases against Caligula that many other romans felt). So I'm using him to answer your question.

The answer is that it's clear that the people are still pretending it is still a republic. A few relevant passages below

When he entered the city, full and absolute power was at once put into his hands by the unanimous consent of the senate and of the mob, which forced its way into the House, and no attention was paid to the wish of Tiberius, who in his will had named his other grandson, still a boy, joint heir with Caligula.

This says a few things. First, it shows that the "Emperor" still didn't command the complete powers that they would in later periods. Since, if he had, it is unlikely that the senate and people would have chosen to ignore Tiberius's wishes to have a joint heir appointed. Second, the fact that the senate (and mob) have this power to hail the new ruler suggests that the powers of the Republic are still in existence, just in a modified state.

It is also telling that Caligula did hold offices prior to becoming "emperor"most notably pontiff, but the very fact that he advanced with "strong commendation" before even assuming the job suggests it was mostly to fill the resume. Likewise, he was then encouraged to begin to see himself as a potential heir to Tiberius

Then appointed augur in place of his brother Drusus, before he was invested with the office he was advanced to that of pontiff, with strong commendation of his dutiful conduct and general character; for since the court was deserted and deprived of its other supports, after Sejanus had been suspected of hostile designs and presently put out of the way, he was little by little encouraged to look forward to the succession.

And lastly, after becoming Emperor, Caligula does pay lip service to maintaining the republic, for he did serve as consul instead of outright ignoring it.

1 He held four consulships, one from the Kalends of July for two months, a second from the Kalends of January for thirty days, a third up to the Ides of January, and the fourth until the seventh day before the Ides of the same month. Of all these only the last two were continuous.

Edit: further reading on Caligula by Sue. can be accessed here http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Suetonius/12Caesars/Caligula*.html

There is a point that I remember an author making (who's name I unfortunately can't remember) but I think it's relevant nonetheless. He made the point that "It would have been clear to everyone living in the republic that the government that they lived in was not the Republic of their grandfathers, but it is arrogant of us to assume that this means that Rome had suddenly transitioned into the Imperial state we associate it with. It is likely that the Roman people would still view their government as a republic, albeit one modified to deal with the changing nature of the roman state."

I'll just add that the concept of "emperor" wasn't really formalized during the Julio-Claudian Dynasty. Augustus basically began acquiring a bunch of titles and powers, which is basically what carried over with the other Julio-Claudians. This began to change under the Flavians. Specifically, in 69 CE, Vespasian had the Senate pass the Lex de imperio Vespasiani. This law conferred powers onto Vespasian as the princeps and helped define what we would consider as the Roman Emperor.

Still, though, under the Flavian and later Nerva-Antonine Dynasty (the Principate together with the Julio-Claudians) the Senate retained a not inconsequential amount of power. Throughout the Crisis of the Third Century, this power was diminished as the Empire was stuck in a issue after issue. Under Diocletian the change from Principate to Dominate (where the Emperor was an emperor as we would conceive of in more modern terms) was mostly complete and the Emperor retained most is not all power, ultimately, in the Empire.