In order to activate the internal TNC, the radio has to be in Data mode. According to the manual, press the FUNC button and while the F icon is illuminated on the display, press the SQL button. The Data mode icon, which looks like a square wave, will illuminate.
With your computer's serial port connected to the DB-9 connector on the back of the radio, launch a terminal emulator program on your computer. While Airmail's dumb terminal program may work, I suggest starting with a plain terminal program like Hyperterm or TeraTerm until you get the hang of communicating with the TNC. The default serial port settings for the TNC are 9600 bps, 8-N-1.
Once you have your terminal program communicating with the TNC, hit the Enter key a few times to see if you get a "cmd:" prompt back from the TNC. At that point, you should be able to send commands, like DISP, to the TNC.
Beyond that, I can't help you much. I have two Alinco DR-135T radios, but I'm using the Argent Data T2-135 TNC. I believe the Alinco EJ-41U TNC uses the same Tasco chip that Kenwood used in the TM-D700. That chip is somewhat limited.