Never have used the beet powder but I have colored with clay. It should be a similar method.
You can do it a couple of ways.
1) If you want a solid color soap, you can just mix it in with your oils and stick blend as usual.
2) If you want a swirl, you can take out the lightly traced soap you want to color, take a bit out of that, stir in your clay, and then stir the clay in to color the rest of the soap. Then you can continue with your swirls, ribbons, or marbling.
Personally, I'm lazy......I use method 1 the most often.
I've used green clay; it made kind of a nice pale grey-green. Got a little bit more towards the grey during cure I think. Made a nice slip for shaving soap though.