How to create custom column with auto-increment characters

I think your best bet is to SELECT out the current packet table, modifying it along the way, and INSERT it into a new table. Once you have completed this operation, you can drop the original table, and then rename the new one to the old one.

INSERT INTO newpacket (id, subid, dollar, packetname, `updated packet`)
SELECT p1.id, p1.subid, p1.dollar, p1.packetname, p2.`updated packet`
FROM packet p1
INNER JOIN
(
    SELECT p.id, p.subid,
        CASE WHEN (SELECT p.id - MIN(t.id) FROM packet t WHERE t.subid = p.subid) > 0
        THEN CONCAT(packetname,
                    CHAR(((SELECT p.id - MIN(t.id) FROM packet t WHERE t.subid = p.subid) + 64) USING utf8))
        ELSE packetname END AS `updated packet`
    FROM packet p
) p2
ON p1.subid = p2.subid AND p1.id = p2.id

You can create a trigger.

create trigger my_trigger before insert on mytable for each row 
begin
DECLARE samecount INT;
  set samecount = ( select count(*) from mytable where packetname = new.packetname );
  if samecount = 0 then
    set new.updated_packet = new.packetname;
  else
    set new.updated_packet = concat(new.packetname,conv(samecount+9,10,36));
  end if;
end;

Before a new row is inserted, it counts how many rows with same packetname exist. When there is one or more, the count+9 is converted to base 36 - it's almost the same as HEX except all the way up to Z. So, if count is 1, it becomes 1+9=10=A. The resulting value is concatenated with packetname. If same rows exceed 37, it will not fail, but will append 10 for 38 instead.

Keep in mind that this isn't exactly auto increment and it could be subject to race conditions, when two users insert the same packetname in exactly the same time, the count query can return the same value for both.

EDIT: Note that this solution for when you'll need to insert new rows into that table afterwards and want them to have updated_packet filled automatically. If you want to also update existing rows, one way is to create a new table with same structure, create that trigger on a new table and then do

insert into newtable(id, subid, dollar, packetname)
select id, subid, dollar, packetname from oldtable