Boredom initially encompassing Greek character in part of UK (5)
I believe the answer is:
ennui
'boredom' is the definition.
(I know that boredom can be written as ennui)
'initially encompassing greek character in part of uk' is the wordplay.
'initially' says to take the initial letters.
'greek character' becomes 'nu' (Greek letter).
'in' is an insertion indicator.
'part of uk' becomes 'NI' (Northern Ireland).
The first letter of 'encompassing' is 'e'.
'nu' inserted inside 'ni' is 'nnui'.
'e'+'nnui'='ENNUI'
(Other definitions for ennui that I've seen before include "World-weariness" , "Effect of boredom" , "Feeling of weariness or boredom" , "boring" , "having nothing to do" .)