Mac – Failing Hard Drive

Use the drive as little as possible. Plug it into an enclosure or simply boot the machine in TDM mode and connect it to another machine.

Pull over only the files you need (to minimize the time you spend accessing the drive).

You may want to do something like, just your ~/Desktop, ~/Documents, ~/Pictures, ~/Library/Mail, and ~/Library/Mail Downloads.

Do either: a) the biggest folders last, again, so that if the drive suddenly dies on you, you got as much stuff off it as possible, or; b) the most important folders first, such as any pictures you don't have backed up.

You could use Apple's OS X Recovery Disk Assistant to create a bootable recovery disk on a USB thumb drive. Since you've backed up to an external disk using Time Machine you could replace the hard drive, boot to the recovery disk on your thumb drive, use the recovery disk to format the hard drive and then reinstall 10.8/Mountain Lion. Once Mountain Lion is installed you will walk through the same setup procedure that occurred when you first bought your Mac. One of the first questions in the setup process is if you would like to restore information from Time Machine, which allows you to connect your Time Machine disk and restore your backup.