If this iMac's problem is a dead display backlight but the machine will still turn on, and if the machine has a firewire port, then you might be able to boot it into Target Disk Mode and access the contents of the hard drive that way.
You will need another machine that also has a firewire port. Hold down the 'T' key while you power up the troubled iMac. Then run a firewire cable from that iMac's firewire port to another Mac that is running and also has a firewire port. The troubled iMac's hard drive will show up on the other Mac's desktop as if it were an external drive (which, in effect, it is).
It doesn't matter if both machines have the same speed firewire ports - either can be 400 or 800 - as long as you have a cable with the correct types of connectors (or adapters) to be able to plug into each machine.
After you have finished accessing the 'external' firewire drive, eject/unmount it from the good machine's desktop and then press the power button on the troubled iMac to power it down.