GarageBand is OK, but to play back mp3 AND record at the same time on iOS without having to use iTunes on your PC/Mac to import the mp3, check out Multitrack DAW. It's a serious 24-track recorder/editor, and well worth the ten bucks:
It works even better with Audiobus (also $10),
but even without that, you can just upload your mp3 into Dropbox or whatever cloud-based app you have that supports "Open In", then open in Multitrack DAW. This puts the mp3 on your pasteboard, where you can paste it into the first track of a new tune, then record yourself on track 2, using whatever audio/mic interface (iRig, etc.) you have or the internal mic (not great).
The editing tools in Multitrack DAW are much more professional than GarageBand, I use it on the iPad for all my live recording now.
No, these are not 99 cent apps, but dead cheap for professional-quality tools, IMO.
Merry Christmas, everyone!