Welcome to the forum.
My upgrade path started with Ps version 2.5. That was the last full version, that I had. All the rest were upgrades from there, though to CS 2.
When I went to install CS 2, I was asked for the CS version. I had both the upgrade disc and the S/N (my S/N's went through a couple of iterations, over the decades). In my case, the CS version disc was inserted, found and the installation went smoothly.
Now, while I have CS 5 (and 5.5 and 6), I never installed CS 5, it was just part of the upgrade path, to get me to CS 6 at the end. With my CS 5 S/N, CS 6 installed fine.
That was what worked for me, and I hope will work for you too. Keep that CS 2 disc handy, as you might need to insert that into the multi-drive, at a point in the CS 5 installation.
The one worry that I would have, with an HDD failure, would be Activation, as you probably did not Deactivate Ps, before that failure. For that, should you be up against your allowed Activations, you will likely want to call Adobe Customer Support, and explain your problem. They should be able to reset the Activations, and help you.
Good luck,
Hunt