Nope. It will become a white dwarf at best. Unfortunately the sun is considered to be an average/small star. Much bigger stars will collapse to become neutron stars and much bigger stars than that will collapse into black holes.
Our Sun is far too small to become a Black Hole - it would need to absorb three or four other stars of the same order of magnitude to be massive enough for a Black Hole. The current thinking is that our perfectly ordinary star will eventually become a red giant - it's not even big enough to do the white dwarf trick.
No, our sun can be described as a yellow dwarf. After it swells to the red giant stage it's outer layers will be blown off into space. Then it will become a white dwarf. It will then burn for billions of years until it will eventually become a huge ball of pure carbon under great gravitational pressure. When it cools it could become a giant trillion, trillion carat diamond the size of earth.