In error, Greek character backing exercise system (7)
I believe the answer is:
pilates
'exercise system' is the definition.
(I've seen this before)
'in error greek character backing' is the wordplay.
'in' is an insertion indicator.
'error' becomes 'slip' (slip is a kind of error).
'greek character' becomes 'eta' (letter from the Greek alphabet).
'backing' is a reversal indicator.
'slip' going around 'eta' is 'setalip'.
'setalip' reversed gives 'PILATES'.
(Other definitions for pilates that I've seen before include "Set of exercises used in holistic medicine" , "regime for improved fitness" , "Ali's pet (anag) - exercise regime" , "Holistic system of physical exercises" , "Exercise system" .)