Family court proceedings admitting bishop (6)
I believe the answer is:
tribal
'family' is the definition.
(both can mean relating to one's family)
'court proceedings admitting bishop' is the wordplay.
'court proceedings' becomes 'trial' (I've seen this before).
'admitting' is an insertion indicator.
'bishop' becomes 'b' (abbreviation used in chess).
'trial' enclosing 'b' is 'TRIBAL'.
(Other definitions for tribal that I've seen before include "Like many primitive societies" , "Relating to a social group" , "Relating to clan" , "Of a traditional social group" , "Like certain social groups" .)