If you want to calculate HOMO/LUMO, you have to use computational methods. For measuring the stability of an electrolyte you have to usw a three electrode setup and slowly sweep the potential to very high and very low regions until the current starts increasing. This is the point, in which you start oxidizing or reducing your electrolyte, respectively.
CV of solid samples can be obtained if you can deposit them on a conducting surface. Then you can use this surface as your working electrode in a conventional potentiostat. It is not very difficult to deposit thin films of most material on ITO and FTO surfaces. If your material can be deposited on carbon paper or tape, that can work as well. It all depends on the properties of the material.
If your material can be obtained as a suspension in a solvent (water or organic), you can perform CV experiments on these solutions just like people do for nanoparticles. Best of luck.
Alternatively, you can mix your solid analyte with carbon paste (use about 5% analyte) and then pack the entire assembly in a hollow electrode case (e.g. an electrode contact and plastic sheath - you can get these from BASi, which also sells the carbon paste). Then you can obtain your CV by using the carbon paste electrode as the working electrode in a standard 3-electrode cell. Just fill it with whatever electrolyte solution you'd normally use.