Ungrateful princess drops the Italian who is dying (5)
I believe the answer is:
goner
'who is dying' is the definition.
(if someone is a goner they are going to die)
'ungrateful princess drops the italian' is the wordplay.
'ungrateful princess' becomes 'goneril' (daughter of King Lear in Shakespeare's play).
'drops' indicates named letters should be taken away.
'the italian' becomes 'il' (the in Italian).
'goneril' with 'il' removed is 'GONER'.
(Other definitions for goner that I've seen before include "No-hoper" , "Dead loss" , "One without hope" , "Person beyond help" , "Doomed person" .)