Dealing with a dying 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.