What does the waveform for a Digital Addressable LED Strip look like?
Maybe you can reverse engineer it, but it would be far easier to get a real datasheet.  Whoever you buy these things from should be able to supply you with the relevant technical specs.  If not, don't buy it.

I took a quick look at the web site for the LED strip, and it seemed there was some source code there.  If the source code is well written, then it shouldn't be hard to figure out what it's doing, then duplicate that on whatever processor you want.

Again though, the first step should be requesting proper technical specs from the seller.