Destructive lout beginning to lurk behind London museum (6)
I believe the answer is:
vandal
'destructive lout' is the definition.
'beginning to lurk behind london museum' is the wordplay.
'beginning to' indicates taking the first letters.
'behind' says to put letters next to each other.
'london museum' becomes 'vanda' (V and A, Victoria and Albert Museum).
The initial letter of 'lurk' is 'l'.
'l' after 'vanda' is 'VANDAL'.
(Other definitions for vandal that I've seen before include "One who maliciously causes damage" , "Someone who destroys or defaces property" , "Person who maliciously damages things" , "Lout causing damage to property" , "Destructive type" .)