Kurt/Nick
The secret is to have a DC PSU with variable voltage. The maximum current needs to be at a level whereby the requirements never exceed the maximum available current. If you do that you reach an equilibrium in the cell whereby the metals are plating out and redissolving and therefore you have no resulting metal.
As you increase the voltage the current draw will increase. Also, and importantly the level of metal in solution affects the current drawn too. I made the mistake of running the cell the first time at far too low a voltage- I was at 2V with a current draw of around 1A. Not a lot happened. I increased the voltage until there was a definite fizzing in the cell and gas was visibly being produced. It is important to note that I was stripping metals from a non acidic leech solution so I was not producing Chlorine gas, which you will produce from any HCl based solution. In order to get the cell operating for the particular solution I was running the voltage was around 10V with a current draw of around 13A.
Do note that dimensions of your cell i.e. diameter and distance from cathode to anode, along with volume of liquid AND saturation of metals in solution all affect the current drawn. Dean explains this far better than me I am afraid, but safe to say you need to adjust for the conditions you are operating within.
My DC power supply runs a max of 30V and 60A - not a cheap piece of kit. Dean regularly runs at 200 amps.