I recently read an article which said the the universe is expanding with a very high velocity, which is even a lot faster than the speed of light. So, doesn't this prove Einstein's theory wrong which says that nothing can go at the speed of light, whereas the expanding universe is even faster than that.
Short answer, no.
Relativity "forbids" massive matter moving at (or faster than) the speed of light within spacetime, but the recession of distant galaxies is due to the expansion of spacetime itself. As an analogy, put some points on a graph drawn on a rubber sheet, then stretch the rubber sheet. The points will move away from each other, without actually changing coordinates.
The accelerating expansion (see "Dark Energy") does indeed cause issues with relativity, it implies that there's an (as yet unobserved) energy field "pushing" on spacetime, or that relativity as a theory is subtly incomplete, but even then this isn't an argument against relativity, which is still a valid and useful theory.
In a nutshell, no.
General relativity says that objects with mass cannot travel faster than the speed of light. Space itself can travel faster than the speed of light because it doesn't have mass. However, space isn't moving, it is stretching. If you imagine the Milky Way and other galaxies are on a coordinate plane, the proportions between the galaxies are the same, but the distance is greater. The diagram below illustrates this.
As for the acceleration of this expansion, some scientists believe this is because the cosmological constant is represented by antigravity force (some think this is represented by dark energy, some think this is an inflaton field of some sort) and as the antigravity force pushes galaxies farther and farther apart it slowly overcomes the gravity between the galaxies, and therefore accelerates.
Hope this helps!