Depending on the type of your nanoparticles, their size, their concentration and of course the matrix, different techniques are available: TEM, DLS, NTA, Single particle ICP-MS, …. Let us know more about your NPs to try and provide you with valuable suggestions.
The characterization depends on what you are interested in. If you don't know the identity of the metals present, electron microscopy with EDS might be a good way to start. It may or may not be expensive depending on the facilities available around you.