You can't put tile directly over vinyl. I would peel up the vinyl with the help of an oscillating tool and scraper blade (fein multimaster, dremel multimax, or the harbor freight knock-off). Then install 1/2 backer board (thinset under it, and screw it down). Ready for tile.

The sheet vinyl should be glued-down to a particle board underlayment. Most times the particle board is 3/8" thick, and it is nailed to the subfloor. You want to get rid of the particle board. Instead of peeling up the vinyl, I use the old skil saw. Set the blade depth just right so it cuts through the particle board but not into the subfloor. Run cuts about 12" apart, both directions, to make a grid. The old floor pops up easily with the help of a small flat prybar.

My prior house has particle board/plywood under tile. I tore up particle board. Messy job but straight forward. I did not think this vinyl would have 2 layers under it. Well have to inspect more closely.