Famous Greek Conservative fell in the flipping Med! (8)
I believe the answer is:
achilles
'famous greek' is the definition.
(I've seen this before)
'conservative fell in the flipping med' is the wordplay.
'conservative' becomes 'C' (UK political abbreviation).
'fell' becomes 'hill' (I've seen this before).
'in' indicates putting letters inside.
'the flipping' is a reversal indicator.
'med' becomes 'sea' (Mediterranean Sea).
'sea' back-to-front is 'aes'.
'c'+'hill'='chill'
'chill' put inside 'aes' is 'ACHILLES'.
(Other definitions for achilles that I've seen before include "Heroic warrior" , "Homeric hero" , "Legendary Greek with bad heel" , "Greek hero (with heel and tendon)" , "Heroic Greek with damaged heel" .)