How do I access a hard drive from a computer where windows is deleted?

Download UNetbootin, use it to download the latest version of Ubuntu and create a LiveUSB from it. Boot using that and copy the files to a USB drive.

Note: If you want to go with a smaller distro, get Damn Small Linux or Puppy Linux instead.

If you have another machine available simply pull the drive out and mount it in an external case of some kind to use from another machine.

I couldn't recommend Trinity Recovery Kit enough. It is essentially Karan's answer (LiveCD of a linux distro) but it is built for this purpose. It has a dead simple interface and within a minute or two you can mount his old HDD, expose it as a windows or NFS network share, and browse through the files at your leisure from another computer to find what needs to be copied. If you don't have another computer you can use the included file browser, midnight commander, to move the files to a USB or similar.

get it here -> http://trinityhome.org/

Some other stuff you probably won't use in this case but nevertheless makes it a very cool utility: You can image the whole disk, run antivirus software to make sure you aren't moving a problem to the new machine, recover your windows admin password (Assuming you don't delete windows first :p), it has data recovery tools (for when the data itself has been damaged) and can be loaded in a myriad of different ways (Broken optical drive? chuck it on a USB. USB carked it? Boot from TRK on another computer over the network) Anyway, I promise I'm not a salesman for this thing or anything, I just think it's awesome on account of it saving my job last week when an SSD fell over.