Unplug your old drive and then find a way to install windows on a usb stick or other drive.
Install it as a trial with no key since your not really gonna use it for long.
-Try the drive on a 2nd computer, just make sure you don't let the bios try to boot from it.
Even a windows XP\Vista machine should be ok.
Once you have it working enough (I think it's safe to skip installing some stuff like say the sound card drivers, ect.) Just with the video and basics working is all you need.
Then turn it off, plug in your old drive. In the bios, ect. make sure your booting from the new windows 7 drive not your old one.
Once it's running you can then see what's on it. Copy and save your files.
Copying the drive's USERS\Computername\AppData and start menu over to a folder on your new drive helps to be able to do a "transplant"
You will have to enable "show hidden files" and system files for your folder options to see the hidden Appdata folder.
-There are more files and stuff you should copy and save for future copy-paste stuff.
-While it will take a bigger drive to do it, it's possible to copy your Steam folder, program folders. Look on Steam on how to save and backup your Steam install and gamesaves, ect. Also Ubisoft, ect.
-It's possible to save your browser's favorites into a backup folder or file too. It can then be later copy-pasted over to a new browser install.
-If you do this right you can do a transplant. This is most useful if you wish to go from 32bit windows to 64bit so you are forced to do a reinstall. Like going from windows XP 32bit to 7 x64 for example.
Google\search online about doing the windows transplant and saving your games, saves, programs, favorites, ect. With some effort it can be done.
You could try to run scandisk, ect. while running windows from your new install on the old one and see it if fixes any bad files. Or at least it will tell you if the drive is failing.
-There are portable browsers too that you can use to copy your saved favorites to, like Waterfox portable, Firefox Portable, Opera portable, they can run from a folder or usb stick, ect.