Duke is heading for criminal record (4)
I believe the answer is:
'record' is the definition.
'duke is heading for criminal' is the wordplay.
'duke' becomes 'd' (abbreviation).
'heading for' indicates taking the first letters.
The initial letter of 'criminal' is 'c'.
(Other definitions for disc that I've seen before include "Circular flat structure in the spine" , "Forty-five" , "Body without limbs" , "Sound recording" , "Layer of cartilege between vertebrae" .)