Tweeter is Greek character with thick hair? (8)
I believe the answer is:
nuthatch
'tweeter' is the definition.
(type of songbird, something which tweets)
'greek character with thick hair?' is the wordplay.
'greek character' becomes 'nu' (letter from the Greek alphabet).
'with' says to put letters next to each other.
'thick hair?' becomes 'thatch' (I've seen this before).
'nu'+'thatch'='NUTHATCH'
'is' acts as a link.
(Other definitions for nuthatch that I've seen before include "Flier" , "[FEATHERED FRIEND]" , "warbler?" , "Bird - hunt chat (anag)" , "Trunk-climbing bird" .)