One's own role in play opening in London (8)
I believe the answer is:
personal
'one's own' is the definition.
'role in play opening in london' is the wordplay.
'role in play' becomes 'persona'.
'opening in' indicates taking the first letters.
The first letter of 'london' is 'l'.
'persona'+'l'='PERSONAL'
(Other definitions for personal that I've seen before include "Concerning the self" , "Intimate - in the flesh" , "Distinctive; individual" , "Pertaining to the self" , "Belonging to an individual" .)