Who was Alexander the Great? Alexander (356-323 BC--Alexandros in Greek) became the king of Macedon, a kingdom in the north of modern-day Greece, at the age of 20. He would turn out to be one of the greatest generals in history. He first led a Greek and Macedonian army against the Greeks' long-time enemy--the Persians--and succeeded in conquering the Persian Empire (which included the land of Israel) and Egypt. His army eventually reached modern Afghanistan and the Indus River valley in India. However, at this point his army rebelled and forced him to return back toward Greece. He died at the age of thirty-three in Babylon.
How did Alexander the Great contribute to the writing of the New Testament? During his campaigns, Alexander not only conquered numerous peoples, he also spread Greek culture--including the Greek language--among those conquered peoples. After Alexander's death, due to the absence of a clear heir to Alexander's throne, a number of his generals split up his empire among themselves. They continued his efforts spread Greek culture, a process referred to as Hellenization (after Hellas, the Greek word for "Greece"). As a consequence, in much of the eastern Mediterranean world, Greek became a widely spoken second language (a "lingua franca") among many people groups with very different first languages. This included many Jews. Therefore, when the time came to compose the various books of what we now know as the New Testament, it made sense to write them in Greek, as this was a language familiar to many Jews and non-Jews (unlike Hebrew and Aramaic, languages spoken by many Jews but generally not by non-Jews).
What role did Alexander play in the writing of one of the first translations of the Old Testament? One of Alexander's generals, Ptolemy I Soter, made himself the ruler of Egypt after Alexander's death. His capital city, Alexandria (named after Alexander), became home for a large number of Jewish immigrants. These Jewish immigrants adopted the Greek language as well as other aspects of Greek culture. As a consequence, the need for a Greek translation of the Jewish Scriptures (the Old Testament in the Christian Bible) arose. This translation became known as the Septuagint, from the Greek word for "seventy," due to the legend that it was authored by seventy Jewish scholars. Work on the Septuagint began during the reign of Ptolemy I's successor, though it was not completed until many years later. It was the Septuagint from which the authors of the New Testament quoted. This accounts for the fact that sometimes in our English translations of the Bible verses from the Old Testament quoted in the New Testament are not worded in exactly way as the same verses in the Old Testament--the English translation of the quoted verses in the New Testament follow the wording in the Septuagint, while the English translation of the original verses in the Old Testament follow the wording in the Hebrew Bible.
In short, it was partially the actions of a pagan Greek general some three centuries before that facilitated the quick spread of the Christian message throughout the Roman world in the first century AD. However, I don't believe this was an accident. Rather, it is a clear example of how God works through the actions of human beings, even those who do not acknowledge Him, to accomplish His purposes. In this, Alexander the Great was rather like Caesar Augustus, the Roman emperor whose decree that a census should be taken resulted in the birth of Jesus in Bethlehem as prophesied (see the note on Luke 2:1 in the NIV Study Bible). We can this happening time and again in the course of human events, which should increase our confidence that God truly is in control of history.