Cool character that's written for star (3,6)
I believe the answer is:
fan letter
'written for star' is the definition.
I don't know anything about this answer so I can't tell whether it can be defined by this definition.
'cool character' is the wordplay.
'cool' becomes 'fan' (I've seen this in other clues).
'character' becomes 'letter' (letter is a kind of character).
'fan'+'letter'='FAN LETTER'
'that's' is the link.
Can you help me to learn more?
(Other definitions for fan letter that I've seen before include "Enthusiastic missive" , "Missive from an admirer" , "What may be sent to celebrity" .)