I understand that the Byzantine's were essentially the continuation of the surviving Eastern Roman Empire during the " dark " ages. However, who made up the empire? More specifically, who made up the capital of Constantinople? Were they Greeks? Turks? Romans? Or whom?
Interesting question, and one that cannot be answered simply, nor with definitive figures. It should also be pointed out that it is not as clear-cut as assigning different races to different regions, that's more of a 20th century phenomenon.
Throughout its history, the Byzantine Empire ranged massively in its territory sizes, and therefore "races" it contained, from the intact Eastern Roman Empire of Constantine in 300 to the city-state that it existed as by the fall of Constantinople in 1453.
During the Byzantine period, peoples of Greek ethnicity and identity were the majority occupying the urban centres of the Empire. We can look to cities such as Alexandria, Antioch, Thessalonica and, of course, Constantinople as the largest concentrations of Greek population and identity. Outside the cities, especially in Asia Minor and the Peloponnese peninsula, Greek ethnicity was in the majority yet the regions were not thoroughly Hellenised, with different languages and religious sects surviving well into the ninth century.
Outside the cities and heartlands of the Empire, such as in the Middle East, there was more of a mix of Greeks, Semites and Assyrians. Despite the spread of the Islamic empire in the 600s, the Byzantine possessions in the Middle East remained heterogeneous in their makeup, centring around the cities of Antioch and Aleppo, in what is now Syria, for centuries on until their losses.
On the Western side of the Byzantine Empire, on the Balkan peninsula, incursions of tribes of Slavs and Avars shook up the ethnic makeup of the Empire in the 7th and 8th centuries. Ethnic Greeks were pushed to the few Byzantine strongholds on the coast, and all of what is now Greece was considered the "land of the Slavs." Subsequent military reconquest, proselytisation of Christianity and Greek culture meant that Slavs became part of the ethnic makeup of Byzantium.
Moreover, Byzantine government often practiced a policy of population transfer, in order to take trouble groups out of their homeland. For example, in 530 Justinian moved 10 000 Armenians to Cyprus, in 941, the entire Banu H'abid tribe moved from their Arab homeland to the Empire and Bulgars were transferred to Byzantium after the destruction of their kingdom by Basil II in the 1010s. In this period of Byzantine dominance in the region between 800 and 1050, the Empire was perhaps at its most heterogeneous, as evidenced by the rebellion of Thomas the Slavonian in 820, within Asia Minor itself, who had an army including Saracens, Indians, Egyptians, Assyrians, Vandals, Alans, and Armenians.
In the next hundred years, the racial group that would ultimately destroy Byzantium arrived on the scene: the Seljuk Turks. By chipping away at Byzantine Asia Minor from the 1050s onwards, the Seljuks made the Empire more homogeneously Greek as Byzantium was only able to maintain strongholds surrounding Constantinople.
By the fall of Constantinople to the Ottoman Turks in 1453, it was little more than a homogeneously Greek city-state of around 5000 inhabitants. Asia Minor was inhabited by Muslim Turkic peoples, as was the Middle East. The Greek heartland of the Balkan peninsula was identifiably Byzantine Greek and would remain so throughout its occupation by the Ottomans until independence in 1832.