American cheese is a combination of Jack and cheddar cheeses that is specifically designed to melt
You just basically described Kraft Singles, the first ingredient of which is cheddar cheese.
Kraft Singles are literally just cheddar cheese, milk, calcium/sodium phosphates, and trace amounts of colorings and preservatives. The phosphates are emulsifiers that allow the cheese and milk to mix, and are what gives it its melty texture. They really aren't any different than the stuff you can get at the deli counter, from an ingredients-list perspective.