Floating, and getting control of floating isn't very hard, but it can take a lot of practice.
If you don't have much body fat, you're already predisposed to sink, but managing your position well, and your breathing will help you overcome low body fat. If you already have high or normal body fat, learning how to control your position will allow you to float even better.
Some things you can do to control your posture in the water:
- Relax. This is the most important, if your panicking and flailing, you'll sink.
- Balance. Learn how to position your body so your weight is balanced (or your leverage is balanced). this includes pushing down with your shoulders to compensate for your heavier torso).
- Practice. Whenever you are in the pool, spend some time trying to float. If you can find a partner, have them help you by adjusting your position while you are attempting to float. This will aid you in teaching your body how to position itself.
As far as breathing goes, breath deeply, this will fill your lungs. The better your lung capacity, the lower your overall body density will be and you will float better.
As far as being able to so something like float and play the violin, my guess is that this particular swimmer has a bit of help from her suit. It looks to me like there is a bit of air trapped in the legs. However, there is a great deal of practice involved even still.
Wanted to address one more thing. If you're actually doing the backstroke (or back crawl), in the style of competitive swimming, then you shouldn't have much trouble floating (this isn't really floating per-se, it's swimming, which is different then simply being buoyant). This is because your motion should be a big part of what is keeping you on the surface, your kick should create plenty of upward motion and should keep you on top of the water.