Champion with bull's-eye, almost? (6)
I believe the answer is:
winner
'champion' is the definition.
(I know that champion can be written as winner)
'with bull's-eye almost?' is the wordplay.
'with' becomes 'w' (abbreviation).
'bull's-eye almost?' becomes 'inner' (the circle outside the bull's-eye on a target).
'w'+'inner'='WINNER'
(Other definitions for winner that I've seen before include "film director" , "Victor" , "One that came first" , "Conqueror" , "Top player" .)