quinnj65, welcome to the forum! In some backup programs, there is a cloning feature that you can use. That is most likely what you're going to have to do, because I doubt that a regular backup loaded on it won't work. It has to be formatted, most drives are "bare drives".
There is one possibility to format it, but I can't guarantee it, is using GParted (or another partitioning tool) to format it to NTFS, then your backup may work. But doing it that way, I can't promise anything.
Do you still have your install disc that you originally done your install with? If the hard drive is dying, most likely 7 was installed on it, after Oct 22 of last year. If so, that will format it for you. You can go through the format process, then back out of the install, and apply your backup to the drive.
Honestly, I believe my first suggestion (cloning) is your best option, unless you want to do a fresh install, in that case, if you have your OS disc and COA #, you're good to go. If you don't know your COA# (your serial number), download Belarc Advisor or Jelly Bean Key Finder to retrieve it.
There may be other suggestions posted later. If you need further help, don't hesitate to post back, there's plenty of help here.
Best of Luck,
Cat