Not much of her character is explored, but it's implied she may be vain and is very sensitive. In Target: April O'Neil, she left the team in a huff because the team forget her birthday. It's unknown what else has caused her to leave, but it's likely her vanity and sensitivity are part of her constant leaves.
In one episode, Leonardo mentions that the princess has left the team more than "twenty times" and always returns, to which Michelangelo has indicated that the show "paradoxes" Donnie's life (although he meant "parallels").
Princess is voiced by Mae Whitman.