Rising Greek character, celebrity to come out (7)
I believe the answer is:
emanate
'come out' is the definition.
(thesaurus)
'rising greek character celebrity' is the wordplay.
'rising' is a reversal indicator (going up in a down clue).
'greek character' becomes 'eta' (letter from the Greek alphabet).
'celebrity' becomes 'name' (I've seen this before).
'eta'+'name'='etaname'
'etaname' back-to-front is 'EMANATE'.
'to' is the link.
(Other definitions for emanate that I've seen before include "Issue or spread out from a source" , "Manatee (anag.)" , "Come out (of source)" , "spring" , "Rise" .)