This is all fairly well explained on various Wiki sites. The following is a quote from:
I marked the relevant section in bold; in essence, the creation of the black soulstone drew all the souls of the evils into the stone, together with the one of Diablo. Leah then frees the souls, turning her into the physical incantation of the prime evil.
Diablo spent two centuries slowly corrupting the Soulstone that
imprisoned him. In time, he was able to extend his influence into the
surrounding area and possess both King Leoric and his archbishop,
Lazarus. The King proved too strong to fully possess, so the Lord of
Terror took his son, Prince Albrecht. Diablo then began to shape an
outpost of Hell within the catacombs that ran beneath the town of
Tristram. By spreading terror into the surrounding countryside, the
Demon was able to attract many heroes who came to cleanse the land of
evil. By the time the strongest of these heroes, Aidan, reached this
goal, though, he had become fully influenced by the power of Diablo.
In his twisted state, Aidan believed that the only way to fully
control the Demon was to plunge Diablo's Soulstone into his own head.
This, of course, was exactly what Diablo had planned all along, as the
Demon now had an even stronger body to use to find his brothers and
complete his ultimate plan...
Diablo eventually possessed Aidan and went to free his two brothers,
Mephisto and Baal. Diablo (as Aidan) met Adria the Witch, who knew
immediately that Aidan was possessed by Diablo, and swore eternal
servitude to Him. Eventually, a second party of adventurers confronted
Diablo and killed him. This time however, his Soulstone was destroyed
in the Hellforge, in an attempt to put an end to his evil permanently.
Unfortunately, the destruction of his Soulstone was not enough to end his evil. Upon the completion of the Black Soulstone by Zoltun
Kulle, the souls of all the slain Evils are drawn to the Stone
including Diablo (the location of souls after the slaying of the Evils
is still unknown). Adria betrayed humanity by helping Diablo possess
Leah, the vessel Adria bore 20 years before, the result of conception
of Adria and Diablo. Now that all evils are in one body, Diablo is
reincarnated as the Prime Evil and had begun his assault in the High
Heavens. Unfortunately for him, Diablo was defeated by the Nephalem
and cast down from Heaven.
While his brothers Baal and Mephisto prefer to play power-mongering
games from the shadows through scores of minions, Diablo seems content
to be the pure destructive power behind their machinations. This is
not to say that Diablo lacks the intelligence of his brothers (as he
rarely speaks), but being the youngest, he somewhat defers to their
direction and suggestions in the course of their scheming.