Boredom of Greek character in eastern part of UK (5)
I believe the answer is:
ennui
'boredom' is the definition.
('ennui' can be a synonym of 'boredom')
'greek character in eastern part of uk' is the wordplay.
'greek character' becomes 'nu' (letter from the Greek alphabet).
'in' indicates putting letters inside.
'eastern' becomes 'e'.
'part of uk' becomes 'ni' (Northern Ireland).
'e'+'ni'='eni'
'nu' put into 'eni' is 'ENNUI'.
'of' is the link.
(Other definitions for ennui that I've seen before include "Boredom or discontent" , "Weariness with life" , "boring" , "Boredom in France" , "What's tedious and instils such listlessness" .)