Greek character, first to get into row, a barbarian! (10)
I believe the answer is:
philistine
'a barbarian' is the definition.
(thesaurus)
'greek character first to get into row' is the wordplay.
'greek character' becomes 'phi' (letter from the Greek alphabet).
'first to' becomes 'ist' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'get into' means one lot of letters goes inside another.
'row' becomes 'line' (I've seen this before).
'ist' going inside 'line' is 'listine'.
'phi'+'listine'='PHILISTINE'
(Other definitions for philistine that I've seen before include "One indifferent to culture" , "Someone indifferent to culture and the arts" , "one who's lowbrow?" , "One hostile to culture" , "Tasteless" .)