Was the New Testament written while Jesus was alive?
Jesus is still alive. Or more accurately: has remained alive since His resurrection and will eternally.
Did you mean was it written before He was crucified? No. It was written as a result of His resurrection. However, most scholars agree that there were, most likely, notes taken by the disciples during the life and ministry of Jesus.
Some of the disciples were literate men; most notably was Matthew who was descendant of a Levite and who was a Tax Collector by trade, prior to following Jesus. The first fact would mean he was educated and able to read Hebrew, so that he could grow up to be a scribe. The second fact shows that Matthew was literate in Latin and Greek, as well as Hebrew.
The manuscript evidence between Matthew, Mark and Luke indicates there was probably a document of Jesus teachings that was kept for the Apostles to refer to – perhaps more than one. However, John also points out that Jesus promised that the Holy Spirit would remind them of the things Jesus had taught.