I was making a 5x5x5 RGB led cube but never finished it because i got bored of soldering LEDs haha.
375 individually wired LEDs is good in theory but an absolute mission to do in a way that doesn't look like a wiring disaster.
I was going to use common anode RGB LEDs and connect all the anodes together for each layer of the cube.
All the Red cathodes for a vertical column should be connected together, ditto with the green and blue.
Using this method you can construct the entire cube without any additional wiring, just soldering the LEDs legs together. You'll probably have to make some jigs so you have consistent spacing between all the LEDs.
You'd have 5 mosfets driving the anode for each layer, and 75 (5 hoz row x 5 hoz col x 3 colours) cathodes that need to be driven with shift registers. You shift all the data into the registers for a given layer, latch the data into the storage registers and turn on the mosfet corresponding to that layer. Then you start shifting the data into the registers ready for the next layer, turn off the previous layer, latch the data, turn on the mosfet for the next layer and so forth.
You can use common cathode LEDs instead if your shift register/driver sources current instead of sinking.
Ideally you display each layer at 20% duty cycle and be able to scan the entire cube upwards of 25 times a second for it to seem like the layers are all lit concurrently. If you are able to scan much faster you can implement PWM for the individual colours and be able to display gradients.
74HC595 shift registers can only reliably provide about 7mA of current on each output, and given that using the above scheme each led can only be driven at 20% duty cycle it won't be incredibly bright. You can add transistors to the outputs to boost the current capability but that is a lot of work when...
.. off the shelf dedicated LED drivers provide much more current per output. I was going to use 6 of these:
http://www.ti.com/product/tlc5925As an added bonus, it is easy to make brightness adjustable for all the LEDs connected to a given driver.
Two drivers for each colour (25 outputs used, 7 outputs unused) means you have individual control of the current for each colour therefore being able to tune the white balance (important for cheap RGB LEDs!).
If the LEDs have good white balance you can get away with just 5 driver ICs.