Animal a Greek character keeps at home (5)
I believe the answer is:
rhino
'animal' is the definition.
(short for rhinoceros)
'a greek character keeps at home' is the wordplay.
'a greek character' becomes 'rho' (letter from the Greek alphabet).
'keeps' means one lot of letters goes inside another (inserted letters are kept inside).
'at home' becomes 'in' ('I'm in' can mean 'I'm at home').
'rho' going around 'in' is 'RHINO'.
(Other definitions for rhino that I've seen before include "thick-skinned creature" , "This beast may make a charge" , "old money?" , "one may charge unwary tourist?" , "old-fashioned bread" .)