Jesus was never called "Jesus Christ" while he was alive, according to the NIV translation. Three of the four Gospels name him "Jesus Christ" in their first chapter, but only as part of the introduction. Elsewhere "Jesus" is his name and "Christ" is used separately.
After Jesus' death and resurrection, however, the phrase "Jesus Christ" comes quickly into common use. It can be found in Acts 2:38, 3:6, 4:10, just to name the earliest examples, as well as by Paul throughout his Epistles.
The word "Christ" is a modified Greek translation of the word "Messiah", and when Biblical characters are quoted as using it in the Gospels it's along the lines of "Jesus, who is called the Christ". The non-surname of "Christ" should be considered a title rather than part of his name.