Grave debts following end of dealings with the Queen (7)
I believe the answer is:
serious
'grave' is the definition.
('serious' can be a synonym of 'grave')
'debts following end of dealings with the queen' is the wordplay.
'debts' becomes 'ious'.
'following' means one lot of letters go next to another.
'end of' suggests the final letters.
'with' is a charade indicator (letters next to each other).
'the queen' becomes 'er' (Elizabeth Regina**).
The last letter of 'dealings' is 's'.
's'+'er'='ser'
'ious' after 'ser' is 'SERIOUS'.
(Other definitions for serious that I've seen before include "Solemn, thoughtful" , "Important -- grave" , "Earnest - important" , "Important - solemn" , "Earnest; vital" .)