Best act following hooligan losing head (5)
I believe the answer is:
outdo
'best' is the definition.
(I know that best can be written as outdo)
'act following hooligan losing head' is the wordplay.
'act' becomes 'do' (to do is to act or execute a task).
'following' means one lot of letters go next to another.
'hooligan' becomes 'lout' (both can mean an aggressive person).
'losing head' means to remove the first letter.
'lout' with its first letter removed is 'out'.
'do' after 'out' is 'OUTDO'.
(Other definitions for outdo that I've seen before include "Defeat" , "Perform better than (someone)" , "Surpass someone at something" , "Get the better of" , "going to top?" .)