Inquire impertinently about love becoming something that strangles one! (5)
I believe the answer is:
noose
'something that strangles one' is the definition.
(a person can be hanged with a noose)
'inquire impertinently about love' is the wordplay.
'inquire impertinently' becomes 'nose' (I can't justify this - if you can you should give a lot more credence to this answer).
'about' means one lot of letters goes inside another.
'love' becomes 'o' (In tennis, 'love' means 'zero').
'nose' placed around 'o' is 'NOOSE'.
'becoming' is the link.
(Other definitions for noose that I've seen before include "Halter loop" , "Hangman's or cowboy's loop" , "Circular loop" , "Loop at the end of a rope" , "Hangman's running loop" .)