Assuming that you are referring to a Cockroft-Walton voltage multiplier, please note that as more stages are added the available energy output significantly descreases.
You would be real lucky if the CM multiplier didn't reduce the output voltage to less than the secondary voltage of the NST!
From my own experiences over the years, the ten stages that you propose is simply not going to work for your Tesla coil. From that Wikipedia article:
In practice, the CW has a number of drawbacks. As the number of stages is increased, the voltages of the higher stages begin to "sag", primarily due to the electrical impedance of the capacitors in the lower stages. And, when supplying an output current, the voltage ripple rapidly increases as the number of stages is increased (this can be corrected with an output filter, but it requires a stack of capacitors in order to withstand the high voltages involved). For these reasons, CW multipliers with large number of stages are used only where relatively low output current is required. The sag can be reduced by increasing the capacitance in the lower stages, and the ripple can by reduced by increasing the frequency of the input and by using a square waveform. By driving the CW from a high-frequency source, such as an inverter, or a combination of an inverter and HV transformer, the overall physical size and weight of the CW power supply can be substantially reduced.