For example, Homer’s Greek character gone travelling (6)
I believe the answer is:
pigeon
'for example homer's' is the definition.
(I know that homer is a type of domestic pigeon)
'greek character gone travelling' is the wordplay.
'greek character' becomes 'pi' (letter from the Greek alphabet).
'travelling' is an anagram indicator.
'gone' anagrammed gives 'geon'.
'pi'+'geon'='PIGEON'
(Other definitions for pigeon that I've seen before include "Business" , "Common city bird" , "Epigon (anag.)" , "Bird of towns and wood" , "Go pine for the bird" .)