King goes round US city to chill out (5)
I believe the answer is:
relax
'chill out' is the definition.
(I've seen this before)
'king goes round us city' is the wordplay.
'king' becomes 'rex' ('king' in Latin).
'goes round' means one lot of letters goes inside another.
'us city' becomes 'LA' (Los Angeles).
'rex' placed around 'la' is 'RELAX'.
'to' acts as a link.
(Other definitions for relax that I've seen before include "Abate" , "Rest" , "Put your feet up" , "Moderate" , "Take one's ease" .)