Yes, I put a 0.1μF ceramic capacitor between V+ and GND of the chip (see Fritzing-Image below), that shouldn't be the problem.
However I'm now just running the LEDs from the 5v connector on the arduino DUE and this seems to work properly.
The next thing I wanted to test was the layer switching using transistors (NPN). I included a transistor to switch the V+ path on or off. From what I've read so far it shouldn't be a problem to switch a 5V circuit using a 3V pin. However, as soon as I connect it like in the picture below, I only get VERY dim LEDs, even without multiplexing or PWM. I measured the current drawn from the Switch pin (yellow cable) and it's just 0.1 mA. That's of course far too low to saturate the transistor. according to the Arduino DUE Pinout this pin should be able to deliver 15 mA. And it surely can, because when I connect it directly to a LED, it draws about 10 mA.
The same thing happens when I connect the base of the transistor to 3.3v directly. Only if I connect the base to 5v, the transistor seems to get saturated. But to be able to control the base, I need to connect it to a 3.3v i/o pin on the DUE.
I'm sure the transistor is the right kind, all the values are below the tolerance. I tried several types that should work according to their datasheets, but none worked properly. I even tried a Darlington Transistor, this one had an even worse result (LED light almost not visible).
Should I try a MOSFET? Sadly I don't have any at the moment but ordered some different types for testing, they should arrive next week.
I'm sure I'm missing something crucial here (as I said, I'm a beginner, and many things, like transistors, are still a bit mysterious to me). Maybe I should just order another Arduino UNO. At least there everything runs on 5V. But I'm quite eager to get this to work on the DUE...
This is my current test setup: