This is not an easy question. Modern Greece is arguably quite easy to define: we would probably say that modern Greece started with the revolution of 1821 that saw Greece become independent from the Ottoman Empire. But what about Ancient Greece? The popular image of Ancient Greece with Spartans and Athenians and philosophers and Alexander the Great and hoplites and so on and so on - that is the Classical period, in the 5th and 4th centuries BC. But it is a snapshot of a continuous ( though of course changing) culture that began emerging in the 8th century (more or less), and would continue without enormous upheaval under Roman rule from the 2nd century BC. Things change substantially in the 4th century AD. The emperor Constantine moved his capital to Byzantium/Constantinople/Istanbul. He also introduced Christianity to the empire, and the worship of the Olympian gods began its decline, eventually outlawed. With the split of the Roman Empire, the eastern empire became the Byzantine Empire (in modern terminology). This was largely Greek speaking and combined elements Roman and Greek tradition and culture. This state of affairs more or less continued until the ottoman conquest of Constantinople in 1453.

So to try and sum up: a definitive answer would say that "Ancient Greece" ended in 1453 and modern Greece began in 1821. A better answer might say instead though that politically the Ancient Greece of city states and Hellenistic kingdoms ended with the Roman conquest of 146BC. Of course the culture of the people did not change overnight, but that was the beginning of this process of romanisation.