By far the most natural way to achieve this is for the singer to perform multiple takes - even if you've only one singer. At it's simplest, you could just double track the vocal part, although I prefer 3 parts - panned centre, left & right. Then repeat this process for each harmony, & mix accordingly.
Melodyne & some of the above methods do work on am single voice, but phase cancellation can be an issue if your not careful with each part's timing & panning. If you can only get a couple of decent takes from the singer, far better to perform any harmony/pitching techniques on the second take to avoid any phase issues. It's not uncommon for a singer to perform each vocal part 5-10 times, then comp (edit) the best takes so that you end up with 3 decent takes.
If you've only only one voice to work with Antares Harmony EFX is a useful alternative.
http://www.antarestech.com/products/harmony_efx.shtml