A start, or an upstart, in debt (5)
I believe the answer is:
debut
'a start' is the definition.
(I know that debut is a type of start)
'an upstart in debt' is the wordplay.
'an upstart' becomes 'u' (start of the word 'up').
'in' means one lot of letters goes inside another.
'u' placed inside 'debt' is 'DEBUT'.
'or' is the link.
(Other definitions for debut that I've seen before include "Appear in public for the first time" , "First appearence as of society girl" , "Start -- launch" , "where one began" , "First appearance, a coming-out" .)