Chap I'm backing with each moving line (7)
I believe the answer is:
michael
'chap' is the definition.
'i'm backing with each moving line' is the wordplay.
'backing' says the letters should be written backwards.
'with' means one lot of letters go next to another.
'moving' indicates anagramming the letters.
'line' becomes 'l' (used when specifying particular lines from a poem).
'im' backwards is 'mi'.
'each' with letters rearranged gives 'chae'.
'mi'+'chae'+'l'='MICHAEL'
(Other definitions for michael that I've seen before include "Man's name" , "George -, pop singer" , "- - Flanders, 18's lyricist" , "American musician (originally one of 5)" , "- Faraday, English scientist" .)