Stuff goes tits up for people who care for themselves (8)
I believe the answer is:
egotists
'for people who care for themselves' is the definition.
(people excessively concerned with themselves)
'stuff goes tits' is the wordplay.
'stuff' is an anagram indicator (I've seen 'stuffed' mean this).
'goes'+'tits'='goestits'
'goestits' is an anagram of 'EGOTISTS'.
'up' is the link.
This may not be right. It may be part of another bit of the clue.
(Other definitions for egotists that I've seen before include "Self-centred self-seekers" , "People with exaggerated notions of their own importance" , "Conceited and self-centered people" , "The selfish" , "They have an exaggerated sense of their own importance" .)