Foremost record of year disposing of duke’s animal book (8)
I believe the answer is:
bestiary
'animal book' is the definition.
(a bestiary is a book about animals)
'foremost record of year disposing of duke's' is the wordplay.
'foremost' becomes 'best' (similar in meaning).
'record of year' becomes 'diary'.
'disposing of' suggests deleting specific letters.
'duke' becomes 'd' (abbreviation).
'diary' with 'd' taken away is 'iary'.
'best'+'iary'='BESTIARY'
(Other definitions for bestiary that I've seen before include "Medieval collection of animal descriptions for sybarite" , "Medieval book of animals for sybarite" , "medieval treatise" , "Allegorical work" , "Medieval collection of descriptions of animals" .)