The first woman to go after a Greek character's produce (7)
I believe the answer is:
achieve
'produce' is the definition.
(thesaurus)
'the first woman to go after a greek character's' is the wordplay.
'the first woman' becomes 'eve' (I've seen this before).
'to go after' is a charade indicator (letters next to each other).
'greek character' becomes 'chi' (letter from the Greek alphabet).
'a'+'chi'='achi'
'eve' put after 'achi' is 'ACHIEVE'.
(Other definitions for achieve that I've seen before include "Produce" , "Reach; be successful" , "Effect" , "Bring to a successful outcome" , "Attain (goal)" .)