Your best option is to use an old CRT VGA display or possibly a very early LCD monitor which had lower resolutions & can sometimes be selected on the monitor itself. You would need to boot from an OS X install DVD so you can access Disk Utility to erase the internal HD.
Another option would be to pull the IDE Hard Drive from the Cube and use and IDE to USB Adapter to connect the HD to another computer. These IDE to USB Adapters are getting harder to find though. I cannot guarantee this will work with recent Macs or Windows PCs.....it should. Here is one IDE to USB Adapter I've used in the past:
https://vantecusa.com/products_detail.php?p_id=90&p_name=IDE/SATA
You can try putting the Cube into Target Disk Mode which can be done without a working display. However, this requires a Firewire connection to another computer.....so this option is probably unlikely unless you have a Firewire cable and an older Mac or PC with a Firewire port.
With any of these options you will want to write zeroes to the whole physical drive (single pass is sufficient). Disk Utility has a "secure erase" option when erasing a drive. It may take a while depending on the size, speed, and health of the Hard Drive.....assuming the Hard Drive even still functions.
Another option is to remove the Hard Drive and physically destroy it. Some people bash it with a sledge hammer, others may drill multiple holes into it. Just be very careful not to hurt yourself.
Or you could take it to a trusted tech or recycler to perform the secure erase for you, or the physical destruction of the HD. Or maybe a tech would have access to an old VGA CRT monitor or a very early LCD panel.
