Flash your Arduino with the Illuminatrix Arduino firmware.
Setup communication with your Arduino. You don't actually need Octoprint at all, you can issue commands directly to the firmware, commands like:
If you're using Octoprint on an Raspberry Pi (my recommendation), then you need to provide a means for the Raspberry Pi to talk to the Arduino. Arduino's with a USB interface make this insanely easy.
If using Octoprint, install the Octoprint Illuminatrix plugin. Follow the steps from the readme on github.
Connect your RGB LEDs to your Arduino. If you want to do a basic test first, you can connect the R, G, and B pins of a single RGB LED to the Arduino pins 3, 5 and 6.
If connecting an RGB LED strip, then you have to figure out how to use pins 3, 5 and 6 to switch on/off a power source to the LEDs as the Arduino can't source much on it's own. I've used the approach from here.
I'm using little 2N3904 NPN transistors, whatever you use, check the datasheet and compare with your LEDs. Obviously if you burn your house down (etc), I'm not responsible.
Issue commands to the firmware either directly or through the Octoprint plugin. Everything should now be working.