Imagine you have a particle accelerator that you can crank up to arbitrarily high energies. Because of General Relativity, the particles get heavier and heavier as you dump more energy into them. Will these particles ever become a black hole from this relativistic mass? If so, what will an observer particle moving parallel to the black hole, just below that speed, see? What I'm getting at is, kinetic energy is relative. So therefore relativistic mass is as well. So would these particles be black holes to some observers but not others?
You can't make a black hole simply by accelerating a particle, but it has been theorized that smashing some together with ridiculously high energies could produce micro black holes. This, as far as I know, has never been known to happen at particle accelerators, simply because they are not powerful enough.
Relativistic mass increase is inertial, not gravitational, so no, you couldn't make a black hole by accelerating a particle. What happens is that the more energy you use to accelerate the particle, the more its inertial mass increases, but the more its inertial mass increases, the harder it is to accelerate. There comes a point near the speed of light when virtually all the energy you pump into the system is manifested as mass increase rather than velocity, and you can't make the particle go any faster. The best idea, if you want a really fast particle with huge amounts of energy, much more than could be produced by an accelerator, is to use a cosmic ray. Cosmic rays are fast protons or atomic nuclei which have been accelerated by natural events such as supernova explosions to much higher energies than you could get in an accelerator.
Cosmic rays have been used for particle experiments, but the snag is that they are random events, so you don't know exactly where to put your particle detectors to intercept the best ones. To give you an idea of how great their energies can sometimes be, the most powerful cosmic ray particles have the kinetic energy of a rifle bullet, yet the atomic nucleus which carries this energy is so minute that it cannot be seen in even the most powerful electron microscope!