Dairy product produced by fine Greek character (4)
I believe the answer is:
feta
'dairy product' is the definition.
'by fine greek character' is the wordplay.
'by' says to put letters next to each other.
'fine' becomes 'f' (abbreviation).
'greek character' becomes 'eta' (letter from the Greek alphabet).
'f' next to 'eta' is 'FETA'.
'produced' acts as a link.
(Other definitions for feta that I've seen before include "10 down cheese" , "Salty goat or sheep milk cheese" , "Greek 8 across" , "Soft Greek cheese" , "Type of cheese" .)