Grave bishop pursuing cat (4)
I believe the answer is:
tomb
'grave' is the definition.
(I know that grave can be written as tomb)
'bishop pursuing cat' is the wordplay.
'bishop' becomes 'B' (chess abbreviation).
'pursuing' means one lot of letters go next to another.
'cat' becomes 'tom' (a tom is a male cat).
'b' after 'tom' is 'TOMB'.
(Other definitions for tomb that I've seen before include "Sepulchral monument" , "Place for the burial of a corpse" , "Crypt or grave" , "remains here" , "grave situation" .)