Written Greek New Testament

The New Testament was written in Greek, so I must be even more cautious when dealing with the meanings of English or Latin terms derived from the Greek biblical texts.  At the time of the New Testament writings, Alexandria had a larger Jewish population than Jerusalem.  Greek was spoken by more Jews than Hebrew.  How did the early followers of Jesus Christ understand themselves and their symbolic activities?  Why did all these early Jewish followers of Jesus write in Greek, instead of Hebrew?