What is the easiest way to put down cement backer board prior to laying ceramic floor tile?

I've done it. And The kind of screws you need are not so specialized. Simple drywall screws. About 5 to 8 bucks for a box depending on their length. Now since you say you have no intention of pulling up the old vinyl flooring, and that the subflooring is plywood, I would seriously reconsider. If you are doing it this way out of sheer laziness, then dont do it at all. Also, why are you using cement backerboard for a floor? This is normally used in areas that will constantly be wet, such as a shower wall, steam room, etc. For the floor, and take it from me as I have done this before, and have worked with pros doing this as well. Rip out the vinyl flooring. Its extra work, but hey, do it right the first time not half... well you get the picture. Then once you have that done, you are left with the bare subfloor, typically Plywood or Particle board. Check to see if it is level first, if this is a new house, check to see if it is level anyways. Now, lay down over lapping layers of roofing paper. They should be overlapped by 4 inches. Nail them in place, dont glue or screw it down. Now that you have a good surface to start with, you can now begin your prep work for laying down your tile. Ok, so now you are probably asking, well, why not just leave the vinyl tile under it all, whats the difference. Well, right now, you may not think that it is, but later on you will. The thickness of that tile, plus the cement backerboard, or roofing paper will not transition decently into your adjacent rooms, and it will not look good. And yes, it will make a big difference. If you have any questions on how to go about laying the tile down, any tips, etc... feel free to ask.

If you want it to last you should thin set AND screw the cement board down. Cement board is not just used in wet areas. It is a dimensionally stable product and properly installed it will keep your tile and grout from cracking and/or coming loose. If you try to scimp on the prep then you will be wasting your time and money. Some installers do nail it down but bare in mind that this will void any warranty. Don't forget to tape the joints too. If the vinyl tiles are down tight and height room to room is not an issue then leave it in place and go over top.