Greek was to go-to language of culture and intellectuals in the Roman world, a bit like Latin was in medieval Europe): most Roman emperors, at least of the High Empire, were fluent in it, especially as the eastern half of the Empire was mostly Greek-speaking. They spoke Latin too, of course, although less and less as the Western part of the Roman Empire decreased in importance until it vanished.

The level of education would have varied from emperor to emperor, especially during the crisis of the 3rd century, when military men from the provinces attained the imperial throne. Many of these wouldn't have been considered educated in this society, and would only have spoken a heavily accented Latin (Septimius Severus is known to have spoken in a very strong North African accent) and their native tongue. Other emperors were very learned and spoke many languages: Claudius was a scholar, and definitely spoke Etruscan, for example, since he wrote a history of Etruria (sadly lost).

Armenia is an interesting example as it was a client kingdom that was disputed between Rome and the Persians for almost all of their history. The kings there would have spoken Armenian, the national language, but also Greek, because of the fact the kingdom had been incorporated in Hellenistic kingdoms several times, and also because it was the an international language.

So much so that the Parthian Arsacids used Greek as the official language of their multiethnic Empire, in which Aramaic was the lingua franca. So Parthian client kings would have spoken at least Greek, Aramaic and whatever their local languages was, but probably not Latin.

​

TL;DR Several languages, more or less depending on the emperor. All spoke Greek.

In short, Roman and Greek were the two prevailing languages throughout the Roman Empire. Rome never imposed a policy to instate Latin as an official language among conquered peoples, so many languages continued to be spoken alongside Latin throughout the course of the empire's history. In many contexts, Latin took the position of an official "legal" written language of the military and administrative courts, while spoken language, depending on an individual's background and familial heritage, would have likely been more representative of a local dialect. In many more informal letters, speakers will integrate Latin words into Greek texts (or vice versa), or simply shift between the languages without and repetition. It can be all but guaranteed that a member of the Roman army that did not speak a local language would have been quite terrible at administering rule over conquered lands. Again, Greek prevailed massively, and there were often accusations made against emperors for addressing courts in Greek, as it was found inappropriate.

In short, you're basically asking where each emperor was from, for if they were born outside of the Italian Peninsula, it can be all but guaranteed that they spoke at least one additional language. Septimius Severus, hailing from Libya, spoke Punic as his first language. Elagabalus, whose family originated in Syria, likely spoke Aramaic or Syriac, although there is no documentation of him publicly addressing an audience in these languages that I can find. As most of the emperors even from distant provinces had familial ties to Italy, it can be hard to determine whether their 'home' language was Latin or a more native dialect. One thing that is for certain, however, is that they always wrote in Latin for official purposes.

The language policy of Rome, as you might glean by this point, was to "softly" coerce conquered subjects into adopting Latin, as it would improve their status in the empire if it was spoken and understood by the public. In 180 BCE, the town of Cumae, which had formerly used Oscan as the language of public life, sent a decree to Rome to ask permission to use Latin in its markets and auctions, which naturally was accepted. Latin was spoken predominantly by a class of ruling elite Roman senators, equestrians, and patricians, so speaking Latin for any individual would have been a sign of prestige and adherence to the Roman state. There are only two trilingual documents recorded throughout the entirety of the history of the Empire, which implies that knowledge Greek is generally expected even in areas where Hebrew, Syriac, Demotic, or Aramaic were more commonplace. Of course there are other exceptions, namely papyri prepared by interpreters in these tertiary languages, but these number much smaller than the amount of documents in Greek, and the status of these languages diminished over time, especially in Egypt.

It seems that most emperors knew Greek to some extent, but by and large they refused to publicly address audiences in Greek, instead opting to use an interpreter in cases where it was necessary. There are some instances recorded where emperor Constantine addresses an individual in Latin and is responded to in Greek, which implies that he can understand the language, but refuses to speak it as to not undermine the political significance of Latin. One humorous account recorded by Valerius details a speech given by Roman ambassadors at Tarentum in 284, delivered in Greek by an individual named Postumius Megellus, which was so poorly translated that the Tarentines considered it to be insulting enough to nearly justify a war between Tarentum and Rome. This implies that many Roman emperors and ambassadors used a translator because they couldn't fluently speak Greek in the first place.

The continued presence of Greek prevailed to such an extent that following the division of the state into the Western Roman Empire and the Eastern (later Byzantine) Roman Empires, Eastern emperors had to fight to keep Latin instated as the official written legal language, as so many emperors, and people in general, were just speaking Greek. This customary usage of Latin prevailed up until emperor Justinian basically gave up on Latin as a whole, mandating in 535 that legal texts and courts could use Greek, a decision that was based upon Greek's newfound use within Christian religious texts and worship.

Sources:

Rochette, Bruno. "Language policies in the Roman Republic and Empire." Blackwell companion to the History of the Latin language (2011): 549-563.

Most of them would've known both Latin and Ancient Greek at least. Latin was the dominant language of Italy and the western provinces, where most of the emperors were born and raised (at least before the 5th century or so when you start getting Eastern Emperors actually from the East), so it would've been their first language. Greek was the dominant language in the East throughout the Roman period, and as it was also the language of high culture, literature and philosophy it would be surprising if most emperors didn't receive some of their education in Greek as well as Latin. That was the norm for Roman aristocrats from at least the second century BC, and Suetonius mentions Greek education for almost all of the emperors he wrote about (the texts are all here, ctrl+f for "Greek" to find all the specific passages: http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Suetonius/12Caesars/home.html)

Any emperors speaking other languages would've been down to individual circumstances, but I expect it's pretty rare. The Syrian boy emperors, Elagabalus and Severus Alexander, may have known Aramaic, particularly Elagabalus as he was the high priest of a local god before becoming emperor. Aurelius Victor (20.8) suggests that Septimius Severus was more at ease in Punic than either Greek or Latin (being from Leptis Magna in North Africa). I think we should treat that with some suspicion, but he clearly knew the language - perhaps his boys, Caracalla and Geta did too.