What is the correct way to find an equilibrium constant for the dissolution of a specific chemical at various temperatures?

There is no one best way. Generally speaking, Keq values would depend upon the method of measuring concentration values of components of reaction at equilibrium. In some cases titrations can give great results while for reactions with visible chromaphores, spectrophotometry may be a good bet. The ultimate goal is to find an instrument that will detect a physical or chemical property of the substance of interest that can be related to concentrations of components at equilibrium.