Daughter in contest beginning to outdo record (5)
I believe the answer is:
video
'record' is the definition.
(I've seen this before)
'daughter in contest beginning to outdo' is the wordplay.
'daughter' becomes 'd' (genealogical abbreviation).
'in' means one lot of letters goes inside another.
'contest' becomes 'vie' (both can mean to rival or fight).
'beginning to' indicates taking the first letters.
The first letter of 'outdo' is 'o'.
'd' placed inside 'vie' is 'vide'.
'vide'+'o'='VIDEO'
(Other definitions for video that I've seen before include "Who killed the radio star? (5)" , "YouTube media" , "Recorded visual image" , "Relating to televised images" , "Transmitter of televised image" .)