Lassitude of Greek character in eastern part of UK (5)
I believe the answer is:
ennui
'lassitude' is the definition.
(both can mean a feeling of weariness)
'greek character in eastern part of uk' is the wordplay.
'greek character' becomes 'nu' (letter from the Greek alphabet).
'in' is an insertion indicator.
'eastern' becomes 'e'.
'part of uk' becomes 'NI' (Northern Ireland).
'e'+'ni'='eni'
'nu' inserted into 'eni' is 'ENNUI'.
'of' is the link.
(Other definitions for ennui that I've seen before include "Lower interest rate" , "tiredness" , "bore" , "Boredom in France" , "Feeling of listlessness" .)