Native of Riga, perhaps — extremely eager character (6)
I believe the answer is:
letter
'character' is the definition.
(I know that letter is a type of character)
'native of riga perhaps extremely eager' is the wordplay.
'native of riga perhaps' becomes 'lett' (Lett can mean a Latvian - Riga is in Latvia).
'extremely' suggests removing the centre (only the extremes of the word are used).
'eager' with its centre removed is 'er'.
'lett'+'er'='LETTER'
(Other definitions for letter that I've seen before include "Character - mail - landlord" , "Character - landlord - epistle" , "he leases" , "Form of correspondence" , "Alphabetic symbol" .)