I don't think LPT is a good way to transfer data - the LPT interface is quickly being removed from all new computers.

When you talk about a hex file, I assume you mean a file that will contain your application. Then the answer how to transfer an application to your processor will to some part depend on what processor you select.

If you run one diode at a time, then each diode will be lit for a very short time. Since the maximum amount of current allowed to send through the diode is limited, you will get a very dim display. With 256 diodes, each diode will only be lit for 0.4% of the time...

If you drive 16 diodes at a time, then you will get a brighter display. But don't expect to be able to drive the diodes with 16 times higher current than if they are lit continuously, so you would still get a display that can't take advantage of the full intensity of the diodes.

If you drive 64 diodes at a time, then each diode would be lit for 25% of the time. Even with a safe 10mA diode current, your display will be clearly visible. If you are careful by either adding a hardware LED watchdog or makes sure your multiplexing never locks up, you may then also decide to step up the diode current and get full intensity from the diodes.

Driving 32 or 64 diodes at a time is obviously a bit much if using one processor pin/diode. So you need external latches.

You may go for one long 64-bit SIPO, where you shift out the data for 64 diodes before switching to the next set of 64 diodes. Or you may go for two 32-bit SIPO or four 16-bit or 8 8-bit.

You will need suitable latch chips. With current control or without, depending on how professional you want your product. You will need transistors for handling the other dimension of the matrix. Depending on selected processor, you may need external memory to get more RAM. And you may need an external EEPROM or similar to store extra text or graphics in.

There are huge amounts of information about multiplexed displays available on this and other forums. Start by reading and learning. Then start reading datasheets and think about suitable components. If you then get stuck, you can come back and ask follow-up questions. Just remember that it is your project, so you are expected to spend time learning.

Addendum: If you are going to communicate using RS232 with your PC, then you will either need a level converter like MAX232 with external caps, or a USB-to-serial communication cable that has 3.3V or 5V (must match the supply voltage for the selected microcontroller) logic levels on one side for direct connection to the microcontrollers UART RX/TX signals.