Getting 320V DC from 12V DC
The original circuitry is a self-resonant system of a transformer with 3 coils and a transistor and some capacitances. This is used because it is simple and cheap.

The good news is that with modern core materials and a little more budget the same self-resonant system can be built at, let's say 35kHz outside of the audible range. And with very good power output. Many phone chargers do something similar, but from high to low. 

I am not very in the mood for many self resonant calculations, but I have in stead used the awesome interwebs to find an instructable that isn't absolutely abominable. Be careful with the soldering of high voltage parts, try to keep large distances:

[Instructable aimed at charging HV capacitors] (http://www.instructables.com/id/DC-DC-HV-Boost-Converter/?ALLSTEPS)

NOTE1: I have not completely read it, only scanned over it and determined it not to be terrible.



EDIT1: From the comments from the above link, I lifted a more structured approach using a Linear Technology Chip. To be more precise: Their datasheet. They work through an entire example to create a high-voltage from a low voltage source:

LT3751 Product Page



NOTE2: These high voltages and especially when put into capacitors can be life threatening.

NOTE3: If the source you built or bought is very strong (i.e. can supply quite a bit of power at the 300VDC side without dropping below 50VDC) you must add a small resistor between the source and the power capacitor. If you do not do this your flash tube may go into continuous conduction (when set to use full flash power / no TTL/FastTTL) on the power supplied by the power circuit and explode in a very small manner of time.