Immortal characters are characters who have either lived for a very, very long time, or keep on getting reincarnated into new bodies, or simply cannot die. These characters have the potential to be tragic and in most cases they are. Sometimes they can be less tragic if the character belongs to a mythological species that lives for a very long time like an elf, however if the character falls in love with or befriends a character from a species who has a life span that is greatly shorter than the other character then it becomes more tragic.
When writing immortal characters it is important to think about the effects time has on their personalities, their behaviour, their bodies, and their memories. They may be more withdrawn than mortal characters because they have outlived everyone they once loved and don’t want to lose anyone else to time. They may be fearful of the passage of time making them impatient despite the fact that they literally have all the time in the world. The idea of eternity may weigh down on them. How do their bodies deal with living for so long? Is there a scientific reason how they keep their bodies the same age forever such as regenerating telomeres? How do their brains work with so many memories?
Imagine about how you would feel if you could live forever. It might make it easier to create such characters.