Q: How do I erase the hard drive of an iMac whose screen doesn't work?

Start the faulty computer in Target Disk Mode and it will appear as a drive on the second computer.  Note Apple has removed the secure erase drive feature in El Capitan so if you are using a computer with that post back for alternatives.

This is basically what  I know. -->

How to use FireWire target disk mode - http://support.apple.com/kb/HT1661 - includes description of hardware and software requirements.

If you can not access the hard drive using TDM, another option would be to remove the hard drive before recycling the iMac.

https://www.ifixit.com/Device/iMac_Intel

Additionally you could put the iMac's hard drive into an external drive enclosure for recovery and/or to maintain your old data.

Before you recycle it, have you tried putting a flashlight to the screen? If you can see the desktop, then it may be just the backlight which is a minor repair.