Greek character in Eastern province shows weariness (5)
I believe the answer is:
ennui
'weariness' is the definition.
(ennui is weariness or boredom)
'greek character in eastern province' is the wordplay.
'greek character' becomes 'nu' (letter from the Greek alphabet).
'in' means one lot of letters goes inside another.
'eastern' becomes 'e'.
'province' becomes 'NI' (short for Northern Ireland, a province of the UK).
'e'+'ni'='eni'
'nu' placed inside 'eni' is 'ENNUI'.
'shows' acts as a link.
(Other definitions for ennui that I've seen before include "Effect of boredom" , "to make one list less when connected" , "Sense of tedium" , "Result of boredom" , "boring" .)