Character keeps climbing to ferret about (6)
I believe the answer is:
rootle
'ferret about' is the definition.
(both can mean to search for something)
'character keeps climbing to' is the wordplay.
'character' becomes 'role' ('role' can be a synonym of 'character'**).
'keeps' is an insertion indicator.
'climbing' is a reversal indicator (in a down clue, letters climb up).
'to' written backwards gives 'ot'.
'role' placed around 'ot' is 'ROOTLE'.
(Other definitions for rootle that I've seen before include "Dig (as pig for truffles)" , "Search the ground as pigs do" , "grub" , "Dig with snout" .)