Record books kept by eastern monarch (5)
I believe the answer is:
enter
'record' is the definition.
(enter can mean to enter into a record)
'books kept by eastern monarch' is the wordplay.
'books' becomes 'NT' (New Testament).
'kept by' indicates putting letters inside.
'eastern' becomes 'e'.
'monarch' becomes 'ER' (abbreviation for Elizabeth Regina, British Queen).
'e'+'er'='eer'
'nt' placed within 'eer' is 'ENTER'.
(Other definitions for enter that I've seen before include "Record - stage direction - come in" , "put in for a race" , "Move into" , "Register (as a competitor)" , "Take part in a contest" .)