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' suggests removing the centre (only the extremes of the word are used).
'greeting' becomes 'hello' (hello is a kind of greeting).
'offbeat' with its middle taken out is 'ot'.
'ot'+'hello'='OTHELLO'
'for' acts as a 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!" .)