Before you begin the time-consuming task of removing the vinyl covering from your basement or mudroom floor, you should ask yourself whether it's really necessary; in many cases, you can lay a new floor directly over old vinyl. If there's no other way, and the flooring is very old, you need to test both the vinyl and the adhesive for asbestos. If the test is positive, removing the vinyl is a job for a pro. Once you're sure it's safe to proceed, get yourself a good pair of knee pads. You're going to need them.
Cut two strips through the vinyl about 6 inches apart, extending from one wall to the opposite one, using a utility knife.
Heat the strip with a heat gun, and as soon as the adhesive is soft enough, work a stiff putty knife under it and pry it away from the floor. As soon as the piece you're prying is long enough to grab onto, pull it while you continue to heat the adhesive with the heat gun.
Remove the rest of the flooring by cutting it into strips, heating the strips and prying and pulling them off the floor.
Use a commercial adhesive stripper to soften and remove the glue from the concrete after you've pulled off the vinyl. The easiest ones to use are water-soluble and contain citric acid, but you may need a solvent-based one to remove some adhesives.
Paint the stripper on the floor with a paintbrush, allow it to work for the time specified on the container, then scrape off the softened adhesive with a paint scraper. Keep a 5-gallon bucket handy to collect the old adhesive and discard it as hazardous waste.
Try prying up a corner of one of the tiles with a putty knife. If it comes up far enough, slip the putty knife underneath it and pry the tile as far as you can. Heat the tile with a heat gun to soften the glue and make the tile easier to pry.
Cut two strips through one of the tiles about 3 or 4 inches apart, using a utility knife, if you can't find any tiles with loose corners. Heat the strip with the heat gun, then work the putty knife underneath it and pry it up. Once you remove the strip, you can work the putty knife under the other parts of the tile in turn and remove the entire tile.
Pry up each tile from the rest of the floor by heating it and prying it as soon as the glue softens.
Clean adhesive residue off the floor with adhesive stripper.