I hate to tell you this, but there's nothing easy about removing glue! Some good news, though: There are options available that are less toxic than traditional glue remover. The most non-toxic out there are citrus-based adhesive removers, like the Tile Mastic and Adhesive Remover at Citrus Depot.
Once the adhesive is removed, you will probably be left with a surface that's less that perfectly smooth. Since vinyl is notorious for show imperfections in the surface underneath, ensure good results by adding a layer of luan plywood onto your exposed surface, as underlayment. Luan plywood is thin - it's less than a quarter inch thich. Attach it with ring nails. They have grooves in them so once they're driven in, they won't come out. Keep in mind, though, that when you nail them in, they pull the surface down - which could lead to indentations and divets that are visible once your vinyl floor settles in. To avoid this, use wood filler on the top of each nail after it's used to attach the luan.
One more tip: If you have a built-in dishwasher, make sure you remove it before doing your floorwork. Otherwise, it could get trapped behind the luan, which will raise your floor above its current height.