What is the history behind New Year's Resolutions?

The tradition of making new year's resolutions dates back to ancient Babylon and Rome. Babylonians would return farm equipment that they had borrowed during the year, in Caesar's time Janus became the symbol for resolutions because he had two faces that could look to the past and into the future.

In 153 B.C.E. January was declared as the beginning of the year by the Romans and named after Janus, the mythical god of beginnings. In 46BC when Julius Caesar established the Julian Calendar. After consultation with an astronomer matching the calendar year with the movement of the sun and marking January 1 as the start of the new year.

Should Christians follow the example of ancient pagan religions? Not according to the Bible. "For what fellowship do righteousness and lawlessness have? Or what sharing does light have with darkness? Further, what harmony is there between Christ and Belial? Or what portion does a faithful person have with an unbeliever? 16 And what agreement does God’s temple have with idols?" (2 Corinthians 6:14-16)

New Year's Resolutions have nothing to do with Christianity.