I like to go down 'under' the existing flooring as much as I can. If it means removing an existing floor down to the subfloor, that's what I'd do.
Different strokes for different folks. Even though it's said that 'encapsulating' asbestos glue/tile (w/o having to remove it) is 'safe', as I said, I - me, wouldn't do it. BUT, that's up to you. For one, if and when you decide to sell the house, you may have to 'disclose' the existance of asbestos in the house. And you know how much THAT little piece of info can 'stir the brain'. At the same time, leaving it in place will raise the floor that much higher in the room. And what that means is that doors may have to be cut, transitions/thresholds into adjoining rooms may be different heights, and you may lose under-cabinet foot/toe room. So, after getting enough feedback, decide if you want to remove the floor.
In EITHER case, you need to lay down a cement backer board. It's to be laid PERPENDICULAR to the top-most layer of underlayment (or subfloor if you go down to it.) Seams should not overlap. You should get instructions on how to install the backerboard. (I can describe it but you might not 'understand' it.) Be sure to check with the manufacturer of the tile for installation instructions too. And get a copy of the warranty. These days, almost any reputable company has a WEB site if the Home Center or Retailer can't help you. READ that info first BEFORE you install. 1) You don't want to have to rip up your new floor down the road (which a recent poster had to do), and 2) you don't want to void a Warranty.
If you need instructions on either the demolition and/or installation, post up again and someone (or I) will post a couple of links. After today, personally, I'll be out until next Wednesday. For now, my best to ya and hope this helps.
PS: The backerboard is a rigid board that, among other things, keeps your flooring stiff so your tiles don't come lose on you.