One looking to oust leader, one from Asian republic (5)
I believe the answer is:
azeri
'from asian republic' is the definition.
(from Azerbaijan)
'one looking to oust leader one' is the wordplay.
'one looking' becomes 'gazer'.
'to oust leader' means to remove the first letter.
'one' becomes 'i' (Roman numeral).
'gazer' with its first letter removed is 'azer'.
'azer'+'i'='AZERI'
(Other definitions for azeri that I've seen before include "National" , "Caucasian" , "Asian" , "Foreign citizen" , "Ethnic group living in Azerbaijan" .)