Cheese of fine character (4)
I believe the answer is:
feta
'cheese' is the definition.
(I know this)
'fine character' is the wordplay.
'fine' becomes 'f' (abbreviation).
'character' becomes 'eta' (letter of the Greek alphabet).
'f'+'eta'='FETA'
'of' acts as a link.
(Other definitions for feta that I've seen before include "Greek 8 across" , "White Greek cheese" , "A ewe's, goat's milk cheese" , "Low-fat cheese" , "Eastern Mediterranean cheese" .)