You don't have to back up your hard drive, but you can.
It's not very likely its the hard drive that is making your MP3's skip.
If the hard drive was failing, you would see errors when you boot your computer up, and it would not even be able to read the files.
I had this problem too. I reformatted my hard drive with Windows on it, but left my other drive with all of my music, alone.
I reinstalled Windows XP. This time with Service Pack 2.
What I noticed is that Windows Media Player would skip my songs. So I use Winamp, and its fine.
What program are you using to burn your CD's? Windows Media Player?
If you are, I recommend using Nero Burning ROM instead. It is much more reliable then WMP.