Animal Greek character adopts at home (5)
I believe the answer is:
rhino
'animal' is the definition.
(short for rhinoceros)
'greek character adopts at home' is the wordplay.
'greek character' becomes 'rho' (letter from the Greek alphabet).
'adopts' indicates putting letters inside.
'at home' becomes 'in' ('I'm in' can mean 'I'm at home').
'rho' placed around 'in' is 'RHINO'.
(Other definitions for rhino that I've seen before include "endangered creature" , "thick-skinned creature" , "Endangered wildlife" , "Massive mammal with one or two horns on its nose, in short" , "Horned jungle beast" .)