A Greek character, woman, to perform (7)
I believe the answer is:
achieve
'perform' is the definition.
(I've seen this before)
'a greek character woman' is the wordplay.
'greek character' becomes 'chi' (letter from the Greek alphabet).
'woman' becomes 'Eve' (Eve is an example).
'a'+'chi'+'eve'='ACHIEVE'
'to' acts as a link.
(Other definitions for achieve that I've seen before include "Bring to a successful outcome" , "Have ice (anag.)" , "'Accomplish, attain (7)'" , "Attain (goal)" , "Finish" .)