Below is a step to drop a private database link.
As sys user create the following procedure under the schema from which database link has to be dropped
CREATE PROCEDURE schema.drop_db_link AS
EXECUTE IMMEDIATE 'drop database link db_link_name';
Now execute the procedure
Now the database link of the schema is dropped.
You can check using the below query
select owner,db_link,to_char(created,'dd-mon-yyyy:hh24:mm:ss') from dba_db_links
Now drop the procedure
drop procedure schema.drop_db_link;
Hope this helps ...