For the audio if you want a free editor then go for Audacity, what you would do with it is record one voice on one track, then record the second voice on another track while you listen to the first voice, then record third, fourth, fifth, however many voices you want to record. Then edit it all together, export it and then bring it into the video editor which brings us to the video portion of this post.
The video is slightly trickier, I'm not entirely sure if there's a free video editor that's out there that allows for multiple video tracks but if there is then hopefully someone can post below and let you know, but if you have a Mac then you can use iMovie which is included. If you're only sticking to free editing software then just go for something like Movie Maker or Pinnacle Videospin and record yourself singing along to your recorded song. However, if you have Sony Vegas or Adobe Premiere or some other software that allows multitracking then one option you could go for is to split the screen into several sections, record yourself singing along to each voice then put each video into each of those sections on the screen, you can do this by scaling down each video and putting it all together which is probably the easiest way.
I haven't gone over this in great detail or even gone over all of the different ways you can do it and I apologise if what I've said is confusing, but if you're unsure about anything at all don't hesitate to ask.