RAM and hard-drive do different things, from the sound of things, you need a new hard-drive, but if the system has under 2gig of RAM you want to upgrade that as well.
From the sound of things in your posts, you should replace the main hard-drive as well and re-install Windows on that. You don't want to keep running a hard-drive that comes up with disk errors all the time. So get 2 drives. A trick to do is find an external full size drive that is on sale, and crack the case open, remove the drive and use that. It's often cheaper than buying an internal drive on it's own. I saved about 50% just a week ago by bying a clearance 2TB external drive and taking out the drive to use as a regular internal. Newegg has some good deals on new external drives at times, check them.
A "backup" to the same drive as your files is useless, if the drive fails, you will lose the backup as well. Buy a second drive, and run your backups to that. I suggest a second internal drive or an eSATA or USB3 external if your computer has those interfaces. USB2 is pretty slow for large file transfer such as a full backup.