Can a black hole explode? In the nature many objects and processes have a critical point (like the more mass an object collects). Logically is that inside black holes the more violent things happen the more matter is inside.  Can they reach a critical point inside when they would explode releasing the trapped inside matter?
 A: Yes, but not the way you are thinking. 
First, before matter reaches the black hole. 
Black holes attract matter. Black holes are small, so in falling matter gets squeezed into a small volume when it gets close. It gets compressed, and compression causes heating. Matter gets so hot that it disassociates into charged particles. 
Furthermore, matter orbits around the black hole instead of falling straight in. As it gets closer to the black hole, it orbits faster and faster like a skater pulling in her arms. Spinning charged particles generate very strong magnetic fields. 
The magnetic fields can produce jets of matters flying away from the black hole. The mechanism is not well understood. See https://en.wikipedia.org/wiki/Astrophysical_jet. Regardless, the forces around a black hole are enourmous, and matter approaches the speed of light. So while a jet may not exactly be an explosion, it is more violent than anything you will find on Earth. 
Here is an image from NASA (Artist's conception)


Second, after matter gets inside, it is generally stuck. But quantum mechanics provides a theoretical way out. 
Particle/antiparticle pairs appear out of nothing and disappear again all the time. Usually they vanish extremely quickly, as required by the uncertainty principle. 
But right at the edge of a black hole, it is possible for one to get sucked in and the other to escape. One particle appeared outside the black hole out of nothing. For things to balance, another particle must have disappeared inside the black hole. So the total change is that the black hole has one particle less, and outside has one particle more. This means that black holes can slowly evaporate. Very slowly. It would take far longer than the age of the universe for this to make a noticeable dent in the size of a typical black hole.
A typical black hole contains at least an entire star. But it is theoretically possible for a black hole to be much smaller. It might contain a mountain. Such a black hole would be microscopic - smaller than an atom. 
For a very small black hole, evaporation is much faster. It would give off enormous numbers of particles. They particles would carry away an enormous amount of energy. As the black hole shrinks, evaporation gets faster and faster. At the end is an enormous explosion, and there is nothing left of the black hole. It has completely evaporated. 
But so far, nobody has found any evidence that a microscopic black hole exists. So far, they are only theoretical predictions for now normal black holes could end, far, far in the future. 
A: 
Can a black hole explode? 

Black holes until the 70s were simply thought to be one of two final states in the evolution of a star, the other being a neutron star. Hence the question of their 'explosion' didn't arise. 
However, after Stephen Hawking posited black hole radiation, he immediately pointed out that this was now possible. However, one has to take their mass into consideration and generally the lifetime of a black hole is so long that they outlast the lifetime of the universe by many, many orders of magnitude. He also pointed out that primordial black holes, created at the time of the Big Bang may have the right mass profile to explode in our cosmological era. However, there has been no consensus as to whether such an observations has been made - which might mean that Hawking was wrong about Black hole evaporation or that cosmologists are wrong about primodorial black holes or simply our technology isn't good enough yet to detect them. 
