To find the Mueller matrix you have to measure the Stokes vector for multiple input polarisation states. You can then use least square fitting to find the Mueller parameters. If you know something about the system it would ideal to include that in your model.
How you measure the Stokes vectors depends on what you know about the system. If there is no retardance you only need to measure the linear polarisation parameters. This can be done by rotating a polariser and taking multiple images with a camera and data fitting. If the system is birefringent you then need to introduce a waveplate and rotate or vary its retardance as well as the polariser rotation. There are commercial cameras/polarimeters that will do this for you.
The waveplates introduce a problem that the light has to monochromatic for them to work. If you need multiple wavelengths it gets complex and you need introduce colour filter or use a spectropolarimeter which usually split the wavelengths using a prism or diffraction grating.