Execute right after criminal's bird (6)
I believe the answer is:
condor
'bird' is the definition.
(condor is a kind of bird)
'execute right after criminal's' is the wordplay.
'execute' becomes 'do' (doing is a kind of executing).
'right' becomes 'r' (common abbreviation).
'after' says to put letters next to each other.
'criminal' becomes 'con' (abbreviation for 'convict').
'do'+'r'='dor'
'dor' after 'con' is 'CONDOR'.
(Other definitions for condor that I've seen before include "Andean bird" , "Large bird of prey" , "South American vulture" , "Large flyer" , "Great bird of the Andes" .)