You need two drives, one external and one for the inside.
Boot up with your DVD and copy the user files, your data, if you can to the external drive you purchased. That's gonna become your backup drive because you NEED A BACKUP. Honestly. I'd also try buying a copy of Tech Tool Pro or Disk Warrior and seeing if they can recover files; it just depends on how valuable that stuff is to you. Since you didn't have a backup, however, I'm guessing it's not that important so you might skip the latter step.
Then install the new internal hard drive; it's a pain if you have an iMac but doable. Check out instructions at ifixit. I like my iMacs, but I think it's inexcusable that the most frequently failing component, the drives, is so inaccessible. It's ironic it's easier to replace on a smaller computer, the laptop. But anyway, replace it, reinstall the data you've managed to recover, and off you go.
Then you can use Time Machine to backup the iMac to the external and not have to endure this process a second time.
Rob