First you need to have sufficiently small particles, so you might have to grind your sample or use a nanoclay. Please verify using the zetasizer the size before you proceed. Next you may have to clean up your sample by washing with ultrapure water, centrifugation, decanting the aqueous layer, resuspending etc. If your sample is clean it has a constant, low conductance, so you can use conductance measurements of your dispersion to see whether cleaning is ready by noticing that the conductance doesn't change anymore.
Zeta potential only makes sense if you either fix pH or ionic strength. I would recommend fixing pH (at an appropriate value for your work) and next vary the ionic strength by the addition of KCl (e.g. 1 mM, 3 mM, 10 mM, 20 mM, 50 mM etc.). As the ionic strength increases you should see a similar decrease in zeta potential as you are depressing the double layer. If your particle size is small enough (z-average < 100 nm), you may also use the z-average from size analysis to see a trend particle size. I have attached an interesting article that describes this topic from a modeling approach:
Chang, F. R. C., & Sposito, G. (1996). The electrical double layer of a disk-shaped clay mineral particle: Effects of electrolyte properties and surface charge density. Journal of Colloid and Interface Science, 178(2), 555-564.