How does a star become a black dwarf?
The white dwarf is a small, dense star and over the course of trillions of years, it shines. But when it runs out of energy, it slowly dims and becomes a black dwarf. Black Dwarfs are not present in this time, as they take trillions of years to form and the Universe is only 13.8 billion years old.
After the White-Dwarf Stage.
A main sequence star when it has consumed all of its hydrogen to helium will rearrange and transform itself into a red-giant Star as its core will shrink and outer layers will expand. The Size of the Sun in its main sequence stage is about 0.1 AU but in its Red-giant state the Sun will be as big as 2 AU.
In the Red-Giant stage, the Sun will burn helium to Carbon and also oxygen and when it is out of these elements it will shed its outer layers into Space forming a planetary nebula with a White-Dwarf at the center, a small extremely dense Star about the size of the Earth.
It is hypothesized that when a White-Dwarf cools in about 100 Billion years or so it will no longer be able to radiate any heat and turn Black-the concept of Black-Dwarfs.