Measuring the surface of both sides of an aspheric lens is difficult. You don't say if both sides are aspheric, or only a single side is aspheric. Whether it is one or both sides, many of the problems are identical.
There are two primary methods that are used in the industry be people who make such lenses. 1) Surface profilometer 2) Aspheric interferometer.
1) A surface profilometer uses a precision stylus to physically trace over the part. Typically two scans are done at 90 degrees to each other, and software fits the result. See, for example, www.mahr.com.
2) Aspheric interferometer - Zygo and others make interferometers that can measure the surface of an asphere. Most of these work by taking multiple measurements, then "stitching" the results together to construct the surface. There are limitations on the asphericity departure that they can measure.
Neither of these methods address possible phase errors within the lens (caused by inhomogeneity or birefringence). To measure those, you need an optical measurement that looks at the final output wavefront. If your lens does not form a good image by itself, you can construct a null lens that, when added to your biasphere, produced a good image. The good image can be tested interferometrically or via incoherent methods (star test, for example). Note you have to make the null lens very well, or you will incorrectly test the asphere.
With any lens, relating the opposite surfaces to each other can be tricky. For a biasphere lens (aspheres on both sides), each surface has an optical axis, which represents a line in space. These two axes can be displaced with respect to each other or not parallel, or some combination thereof. To quantify how the two surfaces relate to each other is likely beyond the scope of your question. It requires measurements on both surfaces that can somehow relate the two measurements in a shared coordinate system. For molded optics, a flange can often be molded into the part that makes this relative measurement easier.