The black in the white/clear baloon came about as a medical laser demonstration to show people how they could do very selective treatments, like in eye-surgery to show people how the laser wouldn't cook their corneas, the lens, or the vitreous humor, but would cook/cut/burn the offending matter back in the retina.
So they'd take a clear balloon with a black balloon inside it and pop the black balloon leaving the clear one unscathed because of it's higher absorption of the laser energy, giving the audience a very quick and visceral understanding of what lasers could do and how less invasive certain procedures were than traditional scalpel procedures. (They didn't have to cut the clear balloon to get the black one, etc.)
Some of the things that will help will be getting a large clear balloon and blowing it up as tautly as possible so it's stretched as thin as possible and won't diffuse the laser too much. Then use a smaller black ballloon so it can be inflated tautly so it will pop well when lased. If you're using two balloons of the same size they will pressurize each other to a greater degree, and the black one on the inside will have to be quite underinflated to fit. And the underinflation will cause it to leak, not pop.