Use a utility knife to cut the vinyl into one foot sections across the length and the width, be careful not to damage the subfloor.
Wedge a putty knife under the corner of one of the sections and pull up the vinyl using the knife to break up any adhesive as you work. Repeat the process over the entire floor.
Once you've got t he flooring up, you can use a mixture of a couple of teaspoons of dish soap and some warm water in a spray bottle and spray it over any remaining adhesive to loosen it up, use a putty knife to scrape it off.
If your floors were put in prior to the 1980's call an asbestos removal company because vinyl flooring manufactured before the 1980's contained asbestos. Now you're all set to add your new flooring. I'm Mrs. FIXIT and it's just that simple!