To answer the second part of your question, it depends on perspective. Byzantine citizens would have viewed themselves as Roman citizens right up until the fall of the empire. Byzantine emperors, too, would have been able to trace the lineage of their office directly back to Augustus and the earliest Roman emperors of the principate. With the death of Constantine XI Palaiologos (1449-1453) during the Ottoman siege of Constantinople, the line of Roman emperors was extinguished once and for all.

But to answer the first part of the question requires an understanding of the complex social and political factors at play during that time period. As a successor state to the Roman Empire, the Byzantines placed a great deal of emphasis on the cultural achievements of their Latin-speaking predecessors. Receiving an education in the Latin classics was considered by Byzantine elites to be in vogue, and being well-versed in the histories of Tacitus or Suetonius, or being able to quote poets such as Virgil or Ovid, would have been looked highly upon.

But, by virtue of being located in what was historically a heavily Hellenized region of the Mediterranean, the Byzantine Empire gradually replaced the traditional Latin lexicon with a Greek one. Ancient Greek classics soon became the academic norm, and the high degree of urbanization in the east allowed the Greek ways of thinking to proliferate, especially after the Western Roman Empire fell and the last vestiges of a truly Latin-centered culture slowly disintegrated. By the later stages of the Byzantine Empire, most of the Roman and Latin cultural traces that were detectable in the early stages of the empire were gone, and although the citizenry did call themselves Roman, in reality they were religiously, culturally, and socially far removed.

Sources: Thomas E. Gregory, A History of Byzantium; Michael Psellus, Fourteen Byzantine Rulers; Anna Komnene, The Alexiad

Edit: I forgot to mention this, but foreign contemporaries of the Byzantine state would have also viewed them as Roman in character. From 1077-1307 the Seljuk Turks established the Sultanate of Rum in Asia Minor, Rum being an Arabic derivative of the word Rome. The gradual changes that to modern historians differentiated the Romans from the Byzantines would have gone unnoticed by contemporaries of the empire, and so for all intents and purposes it was a continuation of the Roman Empire of the past.

In regards to the second half of the question, do you mean by outsiders, or by citizens of the Empire?

There is an entire book regarding this question, Hellenism in Byzantium.

In general, Hellenism was regarded as more closely associated with Greece's pagan past, so it wasn't until Michael Psellos attempted to revive Greek philosophy in the 11th century that Byzantium took more notice.

Keep in mind too, that modern Greek/Hellenic identity arose because of the specific connotations of being "Roman" as "subjugated under the Turks", where as Greek could be seen as from an independent identity from antiquity.