Persephone finally caught by that evil Greek character (5)
I believe the answer is:
theta
'greek character' is the definition.
(letter from the Greek alphabet)
'persephone finally caught by that evil' is the wordplay.
'finally' indicates one should take the final letters.
'caught by' indicates putting letters inside.
'evil' indicates an anagram.
The last letter of 'persephone' is 'e'.
'that' is an anagram of 'thta'.
'e' placed within 'thta' is 'THETA'.
(Other definitions for theta that I've seen before include "Make amends for" , "letter from Athens" , "Greek character" , "8th letter of the Greek alphabet" , "letter to Greeks" .)