Character assassination's beginning to irritate (5)
I believe the answer is:
aitch
'character' is the definition.
(aitch is a kind of character)
'assassination's beginning to irritate' is the wordplay.
'beginning to' suggests taking the first letters.
'irritate' becomes 'itch' (itching is a kind of irritating**).
The first letter of 'assassinations' is 'a'.
'a'+'itch'='AITCH'
(Other definitions for aitch that I've seen before include "bone" , "Letter sound that's often dropped?" , "Eighth letter of the English alphabet" , "the start of hives?" , "Aspirated letter" .)