how did the characters on nightmare before christmas died

Well, in "Nightmare Before Christmas," most of the characters are actually undead or supernatural beings, so they didn't technically die. Jack Skellington, the main character, is a living skeleton. The Mayor is a ghost. Oogie Boogie is a giant, animate bug. The citizens of Halloween Town are all monsters, ghouls, and other creatures. So, in a way, they've always been "dead," but they've never had to experience the process of dying.