All right for a Greek character to have an animal (5)
I believe the answer is:
okapi
'an animal' is the definition.
(I've seen this before)
'all right for a greek character' is the wordplay.
'all right' becomes 'ok' ('ok' can be a synonym of 'all right').
'for' is a charade indicator (letters next to each other).
'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 "Native of Central Africa" , "Animal of the giraffe family (much loved by crossword setters)" , "Giraffe-like mammal" , "Small giraffe" , "Ruminant mammal of Africa" .)