Spread carpenter's glue over the back of a sheet of cement backerboard. Set the sheet in the corner of the floor, pressing it down firmly. Sink 1-inch screws every six inches throughout the board's surface, making sure the screw heads are completely sunk.
Set the next piece of backerboard in the same way, at the end of the first board. Lay additional boards, end to end, cutting the end pieces as necessary with a jigsaw. Don't create any four-way intersections. Cover the entire floor.
Lay fiberglass mesh drywall tape over the seams between the boards. Spread thinset mortar over the tape with a drywall knife, getting it completely smooth. Let the mortar set for a day, then lightly sand it with a drywall hand-sander.
Hold a chalk snap line across the floor, cutting the floor in half, and snap the line. Move the string into the perpendicular direction, so it intersects the first line at the middle of the floor. Adjust the string using a carpenter's square to make sure it's at 90 degrees from the first line. Snap the second line.
Spread tiling adhesive over a few square feet at the intersection of the two lines, using your notched trowel. Set your first four tiles in place at the intersection, butting them tightly together.
Work your way out from the center toward the walls, spreading additional adhesive as needed and laying tiles using the lines are guides. Use your vinyl tile cutter to cut the tiles to fit along the walls.