Brilliance of chapter in climbing story (5)
I believe the answer is:
eclat
'brilliance' is the definition.
(eclat means brilliance or flair)
'chapter in climbing story' is the wordplay.
'chapter' becomes 'c' (abbreviation).
'in' indicates putting letters inside.
'climbing' says the letters should be written backwards (in a down clue, letters climb up).
'story' becomes 'tale' (synonyms).
'tale' backwards is 'elat'.
'c' inserted inside 'elat' is 'ECLAT'.
'of' is the link.
(Other definitions for eclat that I've seen before include "Splash" , "Ostentation" , "Enthusiastic approval" , "Splendour; renown" , "brilliant effort" .)