How to control current and voltage in electrolysis
You can't control both voltage and current electrically:  Your power supply will hold one or the other constant, but the properties of the electrolytic cell will then determine the other one.

So you need to hold all of the chemical parts of the experiment constant:  Have a large ion bath so that the concentration doesn't change much as the plating takes place.  Stir well to keep things mixed. Have large electrodes (i.e. small current density) so that small surface changes don't dominate.  Etc.