I already checked the soldering before I started programming the arduino. Pin 16 wasn't soldered that good, so I wired it again. I checked the other pins by using the "diode" test of my multimeter (one Pin to the soldered Pin, the other to the board. All pins are soldered well).
That's right! By using the Potentiometer, I kan vary the intensitiy of the blocks from very dark to very light. I read of some Problems if the contrast isn't adjusted properly. But I am able to adjust the contrast and tried already all "positions" 
The voltage of GND and Pin 2 is between 5,00V and 5,02V.
GND to Pin 3 is between 0V and 5V regulated by the Potentiometer. The "best legibility" is around 0,35V (the contrast isn't to high, and the blocks are already visible).
There are also no bridges between pins :-/
Slowly I'm getting despaired!
What else can I try to fix this damn problem?