Extremely offbeat greeting for tragic hero (7)
I believe the answer is:
othello
'tragic hero' is the definition.
(I've seen this before)
'extremely offbeat greeting' is the wordplay.
'extremely' says to hollow out the word (remove centre letters) (only the extremes of the word are used).
'greeting' becomes 'hello' (hello is a kind of greeting).
'offbeat' with its centre taken out is 'ot'.
'ot'+'hello'='OTHELLO'
'for' is the link.
(Other definitions for othello that I've seen before include "Hello to Shakespeare's tragic hero" , "Eponymous hero" , "Literary character" , "He killed his wife Desdemona" , "drama!" .)