Greek character in bed brought up issue (5)
I believe the answer is:
topic
'issue' is the definition.
(I know that issue can be written as topic)
'greek character in bed brought up' is the wordplay.
'greek character' becomes 'pi' (letter from the Greek alphabet).
'in' is an insertion indicator.
'bed' becomes 'cot' (cot is a kind of bed).
'brought up' says the letters should be written in reverse (in a down clue, letters go up).
'cot' reversed gives 'toc'.
'pi' inserted within 'toc' is 'TOPIC'.
(Other definitions for topic that I've seen before include "Subject matter of a conversation or discussion" , "Issue (for discussion)" , "Theme (of, eg, a talk)" , "Subject (of speech, etc)" , "Something to discuss" .)