Answer:
First, it is necessary to say that the chances of the Sun becoming a black hole are practically nil, because its nucleus is not big enough to become a black hole.
However, in the imaginary case that happens, since the Sun is a star that revolves around itself, its core would continue rotating, so much so that probably the planets would continue with their orbits with absolute normality (including Earth).
This black hole would have the same mass and, consequently, the same gravity, as the original sun. Therefore the orbiting planets would not notice any difference from the gravitational point of view.
Now, life as known currently, would experience a huge change due to the absence of light (black holes do not emit light). The Earth would be under complete darkness and would have an extreme cold that would end living beings on its surface.