Alright for a Greek character to have animal (5)
I believe the answer is:
okapi
'animal' is the definition.
(thesaurus)
'alright for a greek character' is the wordplay.
'alright' becomes 'ok' ('OK' can be a synonym of 'alright').
'for' means one lot of letters go next to another.
'greek character' becomes 'pi' (letter from the Greek alphabet).
'ok'+'a'+'pi'='OKAPI'
'to have' is the link.
(Other definitions for okapi that I've seen before include "Wild animal" , "browser" , "Small giraffe" , "Ruminant mammal of Africa" , "Native of Central Africa" .)