Conceited bachelor - one initially greeting chief journalist (9)
I believe the answer is:
bigheaded
'conceited' is the definition.
(bigheaded can mean conceited or arrogant)
'bachelor one initially greeting chief journalist' is the wordplay.
'bachelor' becomes 'B' (abbreviation e.g. BA=Bachelor of Arts).
'one' becomes 'i' (Roman numeral).
'initially' suggests taking the first letters.
'chief' becomes 'head' (synonyms).
'journalist' becomes 'ed' (short for editor).
The first letter of 'greeting' is 'g'.
'b'+'i'+'g'+'head'+'ed'='BIGHEADED'
(Another definition for bigheaded that I've seen is " Conceited with large cranium".)