Heiko, I will add to all the above that I dont think there is a lot you can do to make the image you posted have a black background.
As Herbko added earlier, you have to have open water behind your subject. I think, that in the image you attached, the background might be a bit to close to the main subject to get a black background. But a superfast shutter speed woill at least darken it.
Try a high shutter speed and a closed down apeture compination. the smallest opening (biggest number) you can get and still get flash exposure correct. You will get the benefit of greater depth of field and a shallower "exposed" scene.
Lovely thing about digital, you can take your camera underwater and practice a whole dive, try different setting and see how far you can venture from your subject before you lose the flash exposure.
Happy Diving!