Replacing tub with walk-in shower

When building the shower floor, is it appropriate to use only cement board
and nothing else, or do I need to apply a PVC membrane under the cement board?

No cement board on the floor at all (but use it on the walls). Converting from a tub to a shower usually will require changing out the trap and probably more of the drain line from 1.5" to 2", plus, it should be moved to the center of the floor. The floor needs to be sloped prior to putting on the waterproofing, or you'll end up with stagnent water sitting under the tile. Neither tile nor grout is 100% waterproof, thus the need for the waterproofing under it to channel everything to the drain.

You might want to check out www.schluter.com and look at the Kerdi product. They have a couple of premade shower pans specifically for replacing a typical tub; one has the drain centered (more traditional) and the newest one (early this year) has the drain near the end. You still need to make sure you have 2", though.