Greek character concealing source of this disgrace (6)
I believe the answer is:
stigma
'disgrace' is the definition.
(I've seen this before)
'greek character concealing source of this' is the wordplay.
'greek character' becomes 'sigma' (letter from the Greek alphabet).
'concealing' is an insertion indicator.
'source of' indicates taking the first letters.
The first letter of 'this' is 't'.
'sigma' going around 't' is 'STIGMA'.
(Other definitions for stigma that I've seen before include "Mark of shame or social disgrace" , "Badge of shame" , "Blot" , "Symbol of disgrace or infamy" , "A stain as on character" .)