A Greek character, the first woman to win (7)
I believe the answer is:
'to win' is the definition.
(I've seen this before)
'a greek character the first woman' is the wordplay.
'greek character' becomes 'chi' (letter from the Greek alphabet).
'the first woman' becomes 'Eve' (first woman in the bible).
(Other definitions for achieve that I've seen before include "Successfully bring off" , "Successfully conclude" , "Bring to a successful outcome" , "Gain or reach by effort" , "Reach or accomplish" .)