CMC surfaces is a very wide branch of study. I mean that they have been studied from many different perspectives (analytical/PDE's, integrable systems, etc.).
Also the choice for such a text would depend on the level of deepness that you're looking for.
That been said, I can recommend this text. It is formed by the notes taken in classes given by Prof. Dr. K. Große-Brauckmann (TU Darmstadt) for 4th and 5th year students of mathematics. The prerequisites include only a short course in the differential geometry of surfaces. The approach here is mainly analytical. The most basic theorems regarding this topic are explained and proved.
If you are looking from some "starting point" in the topic, this might be your text.