Bond chasing chess player in formal attire (5,3)
I believe the answer is:
black tie
'formal attire' is the definition.
(a black tie is an item of formal attire)
'bond chasing chess player' is the wordplay.
'bond' becomes 'tie' (both can mean a connection).
'chasing' is a charade indicator (letters next to each other).
'chess player' becomes 'black' (one of the two players in a game of chess).
'tie' after 'black' is 'BLACK-TIE'.
'in' acts as a link.
(Other definitions for black tie that I've seen before include "It's worn with a dinner jacket" , "Formal evening wear for men" , "One accompanies DJ" , "expensive party-going habit" , "Accompaniment of 11" .)