The asthenosphere is a part of the Earth's upper mantle. The asthenosphere lies underneath the lithosphere which is the Earth's rigid plates (plate techtonics). In a sense, the lithosphere floats on top of the asthenosphere. It is dense, and consists of some partially molten rock. It is considered a low velocity zone.
The asthenosphere is located anywhere from 60-150 miles beneath the surface of the Earth. Please note that different sources will site different thicknesses, up to 450 miles thick. The asthenosphere will not be as thick in certain areas of the ocean such as ocean ridges.
It is difficult to say how much of the Earth is made up of the asthenosphere because there is not a definite way to know how thick it actually is. It's thickness also varies from area to area so this makes it very difficult as well.
The asthenosphere is a layer of "plastic-like" rock that makes up one of the three major parts of the Earth's interior. It is found roughly 40 miles (70 km) under the surface of the Earth.
The radius of the Earth itself is about 3950 miles (6370 km) from the surface to the center of the planet. The asthenosphere extends from roughly 40 miles under the surface to somewhere around 155 miles under the surface. This means that the asthenosphere makes up about 110 miles of the 3950 mile radius of the Earth. Accordingly, I would say that the asthensphere makes up about 2.5% of the Earth.
The two answers given above are fairly comprehensive, and I have nothing to add. However I would like to point out an apparent discrepancy noted by me in the answer by Pohnpei 397.
It has been stated in the answer that "asthenosphere makes up about 2.5% of the Earth". However if we calculate the ration of volume of asthenosphere and that of earth based on the data given in the post this percentage is nearer to 9.8 percent.
Perhaps the figure of 2.5 percent refers to thickness of asthenosphere as compared to diameter of the earth. However volume of a sphere is proportional to the cube of its radius. As a result volumes of a layer of earth of a given thickness near the surface of earth is much more than volume of layer of same thickness nearer to the center of the earth.