Old playwright bitten by Greek bird (7)
I believe the answer is:
goshawk
'bird' is the definition.
(goshawk is a kind of bird)
'old playwright bitten by greek' is the wordplay.
'old' becomes 'o' (abbreviation).
'playwright' becomes 'shaw' (George Bernard Shaw).
'bitten by' means one lot of letters goes inside another (I've seen this in other clues).
'greek' becomes 'gk'.
'o'+'shaw'='oshaw'
'oshaw' going within 'gk' is 'GOSHAWK'.
(Other definitions for goshawk that I've seen before include "Bird used in falconry" , "Short-winged bird of prey" .)