One way of measuring the permittivity of a material is to shine high energy electrons through a thin slice of it. Starting with high energy electrons and measuring the amount of energy lost in the scattering (if the material is thin enough you can assume that the particle crossed the material scattering at maximum one time), with this you can calculate the imaginary part of permittivity at that energy. Using this data with Kramers–Kronig relations, you can calculate the real part of permittivity (as a function of energy) as well. You can read more about this here
Another way is to use light. At infrared and optical frequencies, a common technique is ellipsometry (here is the wikipedia article, it might help). Dual polarisation interferometry or Dielectric spectroscopy is also used to measure the complex refractive index for very thin films at optical frequencies.