Step 12: Oscillators!
A CW multiplier still requires a moderately high voltage, high frequency input. Obviously this cannot be obtained from a stack of 5,000 AA cells, so it must be generated via some oscillatory witchcraft. The most logical spell would of course be a forward mode flyback converter, preferably one which switches on the zero voltage or zero current crossing point to minimize losses.
In order to obtain 225W at 12kV we need 54mA, and assuming a 100% efficiency we’d likewise need to draw 6.25A from a 36V source. Of course 100% efficiency is unobtainable, but 10A at 36V is still not unreasonable for a hobbyist to supply.
A choice oscillator for this task is the current-fed ZVS Oscillator; an LC resonant zero-voltage switching circuit. Although a Hartley or a Colpitts oscillator could be used, both are not switched under no-load conditions, and would thus burn significant amount of power when the MOSFET travels through its linear region.