Greek character in private blew up (4)
I believe the answer is:
beta
'greek character' is the definition.
(letter from the Greek alphabet)
'in private blew up' is the wordplay.
'in' says the answer is hidden in the clue.
'up' shows that the letters should be reversed in order (in down clue: letters go upwards).
'ateb' is hidden in the letters of 'private blew'.
'ateb' written backwards gives 'BETA'.
(Other definitions for beta that I've seen before include "Greek letter -- second in order of importance" , "software under test" , "Second letter of the Greek alphabet" , "character in Marathon" , "second class grade?" .)