Oracle: Trigger to store username after INSERT into column of same table
First of all, in the row-level trigger body you cannot perform DML against the table which fires the trigger. You will get "mutating table" error .
Secondly, you can't modify the row inside AFTER trigger body, only inside BEFORE .   Assuming CREATE_USER field has proper type and size, the trigger can look like 

CREATE OR REPLACE TRIGGER BI_TABLE_NAME
BEFORE INSERT ON table_name FOR EACH ROW

BEGIN
    :new.CREATE_USER:= USER;
END;


Another option would be not to use trigger at all, but have a column with default value USER : 

create table test_table (test_table_id int not null primary key,
 ..., 
created_by varchar2(30) default user);