The basic reason why Greek was chosen for the New Testament instead of Aramaic or Hebrew was that the writers wished to reach a broad, Gentile (non-Israelite) audience, not just a Jewish audience. The spoken tongue used by both the disciples and Christ was highly likely Aramaic, even though such a Semitic language was not the original one used by the Jews.
Greek was the leading written and spoken language of the eastern Mediterranean world when Rome ruled the world during the New Testament period. Indeed, Greek remained the dominant language, especially in the large cities of Alexandria, Antioch, etc., until after the Arab Muslim conquest, long after the time the Western Roman Empire fell in 476 A.D.
By the time of the New Testament church in the first century A.D., Hellenism had greatly influenced and changed the people and culture of Judea. Greek, not Hebrew, was the commonly used language of Palestine during the rule of the Roman Empire. The ability to speak the Greek language was a needed skill in the Roman world as it was used as the standard way of communicating, carrying out business, and so on.
Greek was written and used during the early church period to communicate between people who grew up in different areas of the world and whose native tongues were quite different. The well-known Jewish historian of the New Testament period, Josephus, stated that the ability to speak Greek was very common not only among the general populous but also among servants and slaves.
Letters of the Greek Alphabet
Some evidence that Christ spoke the Greek language does exist in the New Testament. Jesus utilized at least two Grecian words for love in his well-known post-resurrection conversation with Peter regarding how much he loved him. He used the word agape, which is a deep love and phileo, which is brotherly love toward someone we like (see John 21: 15 - 17). These words have no exact corresponding word either in Old Testament Hebrew or in Aramaic.
Jesus' play on words for "stone" and "rock," while talking with Peter in Matthew 16:18, also uses Grecian words that cannot be replicated in another language.
"And I also say to you that you are Peter (Petros, Strong's Concordance #G4074, which literally means a stone), and on this rock (Petra, Strong's #G4073, which literally means a massive rock) I will build My church . . ." (Matthew 16:18).
Mark 7 records the story of a Gentile woman who requested Jesus heal her young daughter. After being initially rebuffed, she persisted with her request. Jesus responded that he was sent to save and serve Israelites and that, "It isn't right to take the children's food and throw it to the dogs" (Mark 7:27).
The New Testament word Jesus used for 'dogs' was the Greek Kunarion (Strong's Concordance #G2952), commonly used in reference to household pets. He could have, but did not, use the much harsher Hebrew term for dogs commonly used by the Jews in their hatred and contempt for Gentiles.
Additionally, Pontius Pilate could well have questioned Jesus in Greek, since few non-Romans in Judea spoke Latin. It is highly unlikely Pilate would have known the local languages of the "despised" and conquered peoples he ruled in the name of the mighty Roman Empire. Hence Jesus, befitting a Jew raised in 'Galilee of the Gentiles' (Isaiah 9:1), could converse with people of both Jewish and gentile backgrounds.
Since the apostles and evangelists in the early New Testament church wished to reach as large an audience as possible, they chose to write their inspired words in Greek and not Aramaic, Hebrew or some other language. Their choice, given that the Bible is the most popular book ever printed, proved wise indeed.