We are installing new rolled vinyl in an old home. We discovered there is an original layer of vinyl then a layer of plywood (underlayment?)and a second layer of vinyl. How much do we have to remove before putting down the new vinyl?
Just the top layer of vinyl. You could remove more if you need to match floor height on adjacent rooms.
You can put down new vinyl over any amount of layers, the only thing to really consider is the height. Of course if it were my house, I would want to get all the old layers up. But sometimes it's nearly impossible to do that.
Floor leveler was mentioned, which is a good product if you removed a small section, and then decided it's just too hard to continue to remove it. (Floor leveler is like a concrete mix: starts as powder, add water, mix well, spread evenly on floor, and let it dry.) Keep in mind if the old vinyl you plan to go over has a textured pattern, you need to 'skim coat' with the floor leveler (or the old pattern will telegraph through) or simply drop a thin subfloor.
But like I said, keep in mind how high you're raising your floor (especially if you have low ceilings). Also use appropriate screws/staples (making sure they were long enough, so you catch more then just the vinyl below). And remember to "skim coat" your plywood seams and screw holes (staple holes generally don't need to be patched, but if you're not sure, it's better to be safe then sorry - just patch it all!).