If your film is thin enough to still have some electrical contact with the substrate, you can probe the VB-DOS by UPS and the CB-DOS by IPE (inverse photoemission, the machines for that are pretty rare, though).

Alternatively you can try STS (scanning tunneling spectroscopy), but that requires even more conductivity to be left in your film than UPS and IPE.

The answer given by Jürgen Weippert is the best way of characterization to know the electronic structure information. Such characterization directly allow you to probe it. The indirect way of calculating density of states at chemical potential is by measurements of transport properties. If you measure the electrical resistivity and Seebeck coefficient, you can calculate the density of states by using the relationship between transport coefficients and density of states. Note that, most of the time people use equation based on single parabolic band model. Which is very crude way to do. However, you can try in case the band near chemical potential have non degenerate single band. Additionally, an electronic structure calculations can help you to get the desired parameters. At the same time you have to keep in mind that simple DFT calculations do not take the temperature dependent effect in calculating the electronic structure. So the calculations is always corresponding to th absolute zero K. Further, using the electronic structure information, you can also compute the transport coefficients using semi classical Boltzmann transport theory based code, which is based on the constant relaxation time approximation. If you can measure the electrical resistivity very accurately for your sample, you can back calculate the relaxation time, and get the correct electrical conductivity from the calculations. I hope it will help you for your research. Let me know if you need further clarification.