In the past, I have mentioned in just about every forum where the issue has come up, that the most important thing to do when disposing of an old computer is to keep the hard drive.
Sure, if you are trying to sell the old PC it will sell better with the hard disk, but in that case you should first wipe the data off it with a secure wipe program (like DBAN) not just format it.
Back to my suggestion that you keep it.... If you keep your old hard disk you will always have access to it as a point in time backup of what your old system had installed and what files you had at the time of your upgrade. This is handy if you forgot to copy something over to the new system, or you lose a file. Just stick that disk in your handy fire safe (you have one for valuable paper documents right?)
What do you do with it when you want to copy something from it? Get one of these handy dandy USB 2.0 to IDE/SATA adapters. (this link is just an example, search your favorite tech site or computer store and I'm sure you will find one similar to this). You plug one end into the hard disk and plug the included power supply into the hard disk (it should be pretty evident how that works, the connectors only fit one way.) and then plug the USB end of the cable into a USB port on your new PC. It will appear, after a moment, as a new removable drive. ...just like a USB memory stick.
You then can copy and paste files from it, or to it, and use it as your backup hard drive. Maybe make a folder called "OLD PC" and drag and drop all of the current contents into it, then create a folder with today's date "Backup-ddmmmyyyy" and copy your new files that you want backed up from your new computer's C: drive there.