In order to get a new character, something must happen to your current character (if it dies, see Character Death). Characters can wander off and do stuff and may return later. Things to bare in mind-
This is the same as Character Death
Whether your successor character is a descendant, a revenging relative, or a completely different character who hangs around with the party because you misplaced all copies of your character sheet (shame on you) - or you died (one death is a mischance, more than one death - looks like carelessness ;-p)
A new character starts at the same points value (including XP, unspent or otherwise) as your previous character. Minus 30 points. New characters do not start with unspent XP, it becomes spent XP instead. ie if your 250pt+5pt XP+ 10pt unspent XP character dies, you start with a 235pt character.