There is a handy website which shows you how. There are no free windows programs that allow you to go into a dead computer to do this but there is another way to do this, and for nothing..
One presumes the computer does turn on, but just will not install your Windows.
First of all go to the Ubuntu website and download the ISO file containing the Ubuntu operating system. Then burn this onto a CDROM disk.
There is an option to install Ubuntu; DONT USE THAT! The first option is to run Ubuntu direct from the CD. You don´t want Ubuntu loaded onto your computer so this is the right option to use.
Once the Ubuntu desktop appears on your screen you can easily force the hard drive to display its contents for you and you can then back up to an external hard drive. There is an article here which explains exactly how you do this, step by step.