Industry standards allow for vinyl to be laid directly over existing vinyl as long as the existing is the only layer.
To do so the existing vinyl needs to be free of cracking or curling. Because vinyl is very plyable any defects in the floor will telegraph into the new sheet.
To prep the floor you must strip it of any remaining residue. Armstrong makes a good product called "New Beginnings" and can be found in most home improvement stores. Once the floor is stripped, you need to emboss it. Any pattern in the old floor will ultimately show through onto the new pattern. Embossing leveler is put on with the flat side of a trowel. You don't need to put a thick layer. You drag the trowel along the floor leaving only enough to fill in the old pattern. This will dry very quickly, usually within an hour but up to 2.
If the old floor is cracking or curling but is the only layer and not over concrete, you can lay a new 1/4" sub floor. In the past people used luann for sub floor. Most luann today you can buy in Home Depot or Lowes don't meet the quality standards required for a sub floor. Instead you need to buy 1/4" smooth on one side BC pine ply. It is usually close in price to luann.
The plywood needs to be nailed down with a galvanized flat head spiral nail. The seams of the boards and tops of nail heads need to be patched and smoothed over with a high quality wood filler. Once that had dried you will need to lightly sand the areas to make sure the entire floor is very smooth. Any area you have even a minor defect will show through to the new floor.
Assuming you are using a full spreed sheet vinyl, you will want to cut the vinyl before you move it to the room. If this is a bathroom or small kitchen, you can make a pattern using rosin paper. Be careful, cut wrong and it cannot be repaired.
I would always buy the vinyl manufacturer's recommended adhesive. The container will tell you the size trowel you need. This is very important to follow. Trowels ensure you put the right amount of adhesive down. Too much or too little can cause the floor to fail.
Starting and the back of the room, trowel adhesive making semi circles. Work your way out of the room. You need to let the adhesive dries. It will go from a paste like substance to a chewing gum feel. ****Very important to allow this to dry****
Starting at the front of the room you take the vinyl which has now been rolled in reverse. Roll this out into the room. Hopefully you have cut this out in advance.
Make sure once the floor is laid you roll the entire floor with a 100 lb roller to ensure all air bubbles are out. The edges of the floor at the wall and door need trim to make sure they do not curl with time. If you already have wall base 3/4" is shoe or quarter round will do fine. Otherwise you can use new wall base.