Conceited person, British one, initially greeting principal (7)
I believe the answer is:
bighead
'conceited person' is the definition.
(I've seen this before)
'british one initially greeting principal' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'one' becomes 'i' (Roman numeral).
'initially' suggests taking the first letters.
'principal' becomes 'head' (synonyms).
The initial letter of 'greeting' is 'g'.
'b'+'i'+'g'+'head'='BIGHEAD'
(Other definitions for bighead that I've seen before include "Conceited person (informal)" , "Conceited person (usually male)" , "Elder Dumbledore" , "Conceited person with outsize cranium" , "Braggart" .)