this process is dangerous, because like most organic solvents, MEK is flammable, so take fire precautions.
open all the windows, put fans blowing out the windows (and open a door or window somewhere for fresh air to refill the room from), wear gloves, a full face respirator rated for VOC's and isocyurans, then go at it with MEK. just pour the MEK along the edge of the carpet, wait a minute, then pull. the carpet will just lift up. pull until it stops, and repeat. dispose of carpet immediately outdoors, and allow room to air out for 24 hours.
grind surface with a diamond cup wheel. just knock off the high stuff, you don't need to take it down to virgin concrete. just make it smooth.
use a mortar like flextile 52 or ceraflex 610 and install your tiles.