I've found lots of tutorials on how to bleed the disk brakes. And in all of them there is nothing about removing existing fluid. I'm quite afraid that when I start to bleed brakes the existing fluid will be mixed with a new one. Is there any special procedure to remove the existing fluid and clean the cable?
You don't need to remove existing fluid prior to bleeding. Unless you use two incompatible fluid types (for example replacing DOT 5.1 with 5), mixing an existing fluid with new one is completely normal and part of standard procedure. The only situation where you may want to remove the old fluid completely is if you know it has ben heavily polluted with water. But even then normal bleeding procedure is enough to have satisfactory results.
During a standard bleeding procedure you push the old fluid with a new one and they don't have even enough space to mix together. If you do want to remove the fluid before, just push the old one with air instead of new fluid. It may take a few attempts to get rid of everything, but the result should be satisfactory.
And if you're going to fill the system with an incompatible fluid, such procedure is not enough anyway. In this case you would have to take the brake apart, clean everything with alcohol, let it dry out and put back together (paying particular attention to rubber parts). Also note, sometimes a given brake won't accept other kinds of fluids because of its design. In some cases o-rings can be damaged if wrong fluid is used, even if they are clean and dry.