Record verses about one’s part in serial (7)
I believe the answer is:
episode
'part in serial' is the definition.
(eg part of a TV serial)
'record verses about one's' is the wordplay.
'record' becomes 'EP' (abbreviation for extended play record).
'verses' becomes 'ode' (ode is a type of poem).
'about' means one lot of letters go next to another.
'one's' becomes 'I's' (Roman numeral).
'ode' after 'is' is 'isode'.
'ep'+'isode'='EPISODE'
(Other definitions for episode that I've seen before include "Set of incidents" , "Incident or event as part of sequence" , "Part of TV series" , "Incident forming part of a sequence" , "Instalment of a TV series" .)