The Phantom Reaper is the self-appointed supervillain-name of Jack Napier. After he survived a fight with The Shady Avenger at the end of the movie by sheer luck, he managed to jump out of a crashing helicopter just before it hit the water, Jack decided to let go of the Masked Marauder, his identity in that movie, and take on a new role.
He underwent some plastic surgery to get his burned face restored and replaced his old, bloody mask (which reminded too strongly of Jason Vorhees of Friday the 13th fame) by one depicting a skull, generally assuming a skull-themed outfit (his helmet when driving bikes is also black with a skull). The costume makes the name - so he went for “the Phantom Reaper”.