Disgrace, having no time for Greek character (5)
I believe the answer is:
sigma
'greek character' is the definition.
(letter from the Greek alphabet)
'disgrace having no time' is the wordplay.
'disgrace' becomes 'stigma' (I've seen this before).
'having no' suggests deleting specific letters.
'time' becomes 't'.
'stigma' with 't' taken away is 'SIGMA'.
'for' is the link.
(Other definitions for sigma that I've seen before include "How Socrates ended" , "Greek letter signifying sum in maths" , "Letter from Greece" , "local opening in Salonika" , "Greek letter corresponding to S" .)