Cheese produced by loud Greek character (4)
I believe the answer is:
feta
'cheese' is the definition.
(type of cheese)
'by loud greek character' is the wordplay.
'by' means one lot of letters go next to another.
'loud' becomes 'f' (short for forte - 'loud' in music).
'greek character' becomes 'eta' (letter from the Greek alphabet).
'f' next to 'eta' is 'FETA'.
'produced' is the link.
(Other definitions for feta that I've seen before include "A ewe's, goat's milk cheese" , "Low-fat cheese" , "food there" , "White Greek cheese" , "Greek 8 across" .)