Ryan -
your schematic is problematic in several places:
The charging switch won't work the way suggested; when enabled, R5 will conduct a current up to 0.4A, thus dissipating 160W. Q2 needs to be capable of blocking the full charging voltage. If it's got to be an external charge voltage supply, better use an NPN power transistor (or even still better a darlington) in common-collector circuit, place a resistor between base and collector and use another (smaller) high voltage transistor to pull the base low when charging should be disabled.
Yet, my prefered approach would be to directly make a flyback converter (depending on the time you've got to charge the capacitors, a simple inverter with a single inductor and a fast MOSFET switch will do the trick). You could power the whole thing via a 24V DC supply and use a feedback line to disable the converter when the required voltage has been reached.
You may want to place bleeder resistors across each of the charge storage capacitors to even out the voltage distribution.
Your relay won't survive the torture of dumping the charge into the coil for long. The contacts of the 16A relay will weld closed within a few shots. Also contact bounce is a very bad thing here. That's a typical job for a "hockey puck" SCR (Thyristor) which can be found second hand relatively inexpensively and comes rated for impulse currents of several tens of thousands of amps...
I've still got a similar (unfinished) project in a drawer that's been designed to magnetize neodymium magnets after installing the "blanks" inside a hollow shaft, yet the power is much higher, using a
3200µF 1kV charge storage cap... Even though the project that this contraption was supposed to be used for eventually came to a stop (high speed turbo generator), I may still build it, just for fun and since the compopnens are already "collecting dust" in the basement.

Most important when "playing" with such voltages / energies, always be safe and don't electrocute yourself!