Doom to go after family secret (11)
I believe the answer is:
clandestine
'secret' is the definition.
(synonyms)
'doom to go after family' is the wordplay.
'doom' becomes 'destine' (synonyms).
'to go after' says to put letters next to each other.
'family' becomes 'clan' (clan can mean a large family).
'destine' put after 'clan' is 'CLANDESTINE'.
(Other definitions for clandestine that I've seen before include "backroom" , "Surreptitious" , "Concealed" , "Done secretly" , "Private" .)