Boredom shown by Greek character in key part of UK (5)
I believe the answer is:
ennui
'boredom' is the definition.
(I know that boredom can be written as ennui)
'greek character in key part of uk' is the wordplay.
'greek character' becomes 'nu' (letter from the Greek alphabet).
'in' is an insertion indicator.
'key' becomes 'e' (musical key).
'part of uk' becomes 'NI' (Northern Ireland).
'e'+'ni'='eni'
'nu' going inside 'eni' is 'ENNUI'.
'shown by' acts as a link.
(Other definitions for ennui that I've seen before include "Low interest rate" , "Listless feeling" , "bore" , "boring" , "Weariness with life" .)