Towards its end, we know that Byzantine Empire was more or less Greek, while it continued to call itself the Roman Empire (please correct me if it's not so).
What was the phase in its history, when it became more Greek than anything else, in terms of language, customs, and ruling class?
An auxiliary question is please suggest me some good sources about the social and cultural life in the empire.
My understanding of the Eastern Empire is that the Greeks never really lost their identity.
Even when Rome was ruling the area directly, Greek remained the lingua franca in the East - and indeed the official one for all practical intents if not legally. There were a few efforts to enforce Latin as the official language but they never got anywhere:
The emperor Claudius tried to limit the use of Greek, and on occasion revoked the citizenship of those who lacked Latin. Even in addressing the Roman Senate, however, he drew on his own bilingualism in communicating with Greek-speaking ambassadors. Suetonius quotes him as referring to "our two languages," and the employment of two imperial secretaries, one for Greek and one Latin, dates to his reign.
The everyday interpenetration of the two languages is indicated by bilingual inscriptions, which sometimes even switch back and forth between Greek and Latin. The epitaph of a Greek-speaking soldier, for instance, might be written primarily in Greek, with his rank and unit in the Roman army expressed in Latin.
In the Eastern empire, laws and official documents were regularly translated into Greek from Latin. Both languages were in active use by government officials and the Church during the 5th century. From the 6th century, Greek culture was studied in the West almost exclusively through Latin translation. Latin loanwords appear liberally in Greek texts on technical topics from late antiquity and the Byzantine period.
If you really want to put a date on when some kind of transition occurred whereby it reverted to being purely Greek, the collapse of the Western Empire might fit the bill. So might the year when the Byzantine Empire switched from Latin to Greek as its official language (610 or thereabouts - see MAGolding's comment below). But this should be with the understanding that, prior to that, it was very Greek already.
This really only supplements Dennis De Bernady's answer. The Eastern half of the Roman Empire, which we call Byzantium after the former name of its capital city, had previously been part of the Greek/Macedonian ruled Empire of Alexander the Great, and the successor states into which it split after his death founded by his generals e.g. Ptolemy in Egypt. Hence Greek was the language of the rulers and became the 'lingua franca' of trade and cultural exchange. When the Romans took over the region, as government and commerce were already mostly conducted in Greek and this seemed to work, and educated Romans already tended to know Greek, there was no reason to change it.
When I have visited former parts of the East Roman Empire like Greece, Turkey and Egypt where ancient inscriptions remain from the Roman period they are very, very rarely in Latin, but almost always either in Greek or a local language.
The New Testament was written in Greek as the main literary language of the region even though it was then part of the Roman Empire, Jesus preached in Aramaic and the scriptures he quoted were in Hebrew.
When the Western Latin-speaking half of the Roman Empire fell to the barbarians in the fifth century, and the attempt at partial reconquest under Emperor Justinian in the Sixth Century did not last, most of the Latin speaking parts of the Empire were gone, and the Greek speaking parts continued speaking Greek, so Latin fell out of use.
This was not 100%; speaking of Latin in the formerly barbarian provinces of the northern Balkans had been encouraged and survives today in the Latin-derived language Romanian (whose name is related to Roman) and until the nineteenth century in the now extinct Dalmatian language.
Also, when Roman law was codified under Justinian in the sixth century it was done (like the earlier laws on which it was based) in Latin, producing the 'Digest' of laws which is the main form in which Roman law was studied in Europe thereafter.
The Roman Empire became Greek only partially. Asia Minor, Greek lands and the majority of towns became Greek. However, in the Balkans, in the countryside, all Romans and later their heirs, the Vlachs (Romanians and Aromanians), kept the Latin language.
The pagan nomad Slavs occupied parts of Balkans, destroyed all of the towns and, due to their ferocity, forced the Vlachs to flee into the forests and mountains.
After the Slavs Christianization, a part of the Vlachs were Slavicized. They attended the Slavic churches and got Slavic names.
Balkan history was dynamic in changes but Latinity still exists until today through the Aromanians who lived and live in all Balkan states, and through the Romanians.
The Byzantine Empire was always a Hellenic oriented empire...from its earliest foundations dating to Constantine and Helena, to the final days of the Paleologos Dynasty. For over 1000 years, Constantinople, was the Capital of a Medieval Greco-Christian Empire.
With regard to the Roman nature of the Byzantine Empire, it is true that during the first few centuries, the Byzantine Empire was governed by a mixture of Greek and Roman Emperors-(there were even a few Illyrian Emperors, as well as Emperors who of mixed Illyrian ethnic descent, such as Constantine and Justinian). While Latin was communicated throughout the Administrative, Juridical and Aristocratic classes during the Empire's early years-(i.e. early 300's-600 AD/CE), the Byzantine Empire's association and identification with the Greek language, as well as with Greece and Asia Minor and the legacy of Hellenism..... never really disappeared.
One of the more distinguishing characteristics of Byzantium's Greek identity-(from its earliest beginnings), was also its own uniquely distinct version and practice of the Christian religion. The Byzantines practiced an Eastern type-(or rite) of Christianity in a variety of ways. The Supreme Religious Leader of The Eastern Rite/Byzantine Church, was and is....The Patriarch of Constantinople-(who is viewed as the Heir to Saint Andrew). The Byzantine Empire had and still has...Middle Eastern based Patriarchates in Jerusalem and Alexandria, Egypt. Theologically speaking, the Byzantine Liturgy, as well as its style and tone of hymn singing differs from the Latin mass. The Byzantine style of ecclesiastical architecture places a heavy emphasis on iconography-(as well as the simultaneous forbiddance of statuary imagery). But with regard to its Hellenic nature, the Byzantine Church routinely used the Greek language during Church services and the Byzantine Empire pioneered some of the earliest Monasteries in the Middle East and Greece proper. Intellectually speaking, the Byzantines, meticulously preserved and commented on their centuries old textual "Greek Classics" at its Library and University in Constantinople...(which were destroyed by Papal directed Latin Crusader sieges during The Late Middle Ages).
But what truly distinguishes the Byzantine Church from the Western Church, was its emphasis on the mystical, transcendental and spiritual nature of Jesus Christ...from the Anunciation, to the Resurrection and Ascendancy.
These theological, religious, intellectual and historical characteristics of the Byzantine Christian civilization did not start at some random point in the Middle Ages.....they existed from the earliest years of the Empire.
Now, it should be noted that politically speaking, during the first decade of the 600's AD/CE, the Emperor Heraclius-(if my memory is correct), essentially removed all Latin and Roman remnants of the Byzantine Bureaucracy and Culture and began a widespread system of Hellenization throughout the Empire, thereby identifying, associating and distinguishing The Byzantine Empire, as a Greek and Eastern rite Empire which was noticeably separate and distinguishable from its Papal neighbor in Rome. It was from the 600's onward that the Byzantine Empire became-(both territorially and culturally), an increasingly Eastern Christian empire with key geopolitical interests in lands to the North, and especially to the East of Constantinople-(as well as Greece proper).
It is true that the Byzantines traveled westward and had occupied a sizable portion of the Eastern Italian coast, as well as much of Sicily and it is also true that the Byzantines had traveled southward whereby they occupied parts of North Africa. However, these territorial occupations existed during the early years of the empire and due to the Islamic conquest of North Africa, as well as the rise of Papal power within Italy proper, the Greco-Byzantine Christians, from the 600's onward, looked increasingly Northward and especially Eastward.
