Greek hero is sick, gripped by pains (8)
I believe the answer is:
achilles
'greek hero' is the definition.
(I've seen this before)
'sick gripped by pains' is the wordplay.
'sick' becomes 'ill' ('ill' can be a synonym of 'sick').
'gripped by' indicates putting letters inside.
'pains' becomes 'aches' (ache is a kind of pain).
'ill' placed within 'aches' is 'ACHILLES'.
'is' acts as a link.
(Other definitions for achilles that I've seen before include "Famous Greek" , "Greek warrior (with vulnerable heel)" , "His heel was his weak point" , "Ancient Greek with weak heel" , "Greek hero of the Trojan War" .)