Lucifer

"As he fell from grace, he nonetheless fought back in all his defiant fury."

- Lore

Lucifer is the Greater-Scope Villain or Big Bad of most varieties of Christianity, Judaism and Islam (and by extension of most settings in which he is confirmed to exist), and a fair amount of fiction with religious influences, and other religions have similar variants. Traditionally, he is the single most evil being in existence, period.

Christian Mythology
Lucifer's backstory casts him as a Fallen Angel, originally the light-bearing archangel of Heaven, who is damned to spend eternity in Hell due to his rebellion against God. By virtue of his once-grand celestial position, Lucifer is depicted as the king of Hell and the creator of Demons, even if he's as much a prisoner there as any of the damned. This backstory is derived from the Book of Revelation.

No matter what version of Lucifer is employed, he is usually a shapeshifter, capable of taking any form—physical or not—that he cares to. In human form, he is often shown with red or blond hair. His true form is often that of a Big Red Devil. All things that hide or distort the truth are in his province, making him also the master of deception and illusion. Consequently, his most common incarnations are wheeler-dealers, bargainers, and con-men.

Despite an overwhelming overconfidence (he represents the Deadly Sin of Pride, after all), he almost always possesses a keen sense of enlightened self-interest. In keeping with the belief of the "most beautiful angel" who fell, Lucifer takes a handsome face whenever possible, but behind it is often rage and hatred; tearing away that mask, either intentionally or not, is possibly one of the most suicidal things a character can do.

In his more modern incarnations, he is often a charming man of wealth and taste with a taste for the offerings of humanity's cultural achievements, from wine to music to poetry and stories. His interpretations do not always accord with the official ones. Despite this however, Lucifer is usually shown—for all his enjoyment of the finer things—to have a pathological hatred of humanity, blaming them for stealing the rightful place of the angels as God's children.

Hades
Though Lucifer himself doesn't physically appear in the game, his legendary weapon the Adamant Rail is seen and can be used by Zagreus, implying that he may've already been cast out at some point prior to the events of the game.

Zeus himself apparently confirms this, as upon first seeing the weapon he mentions that Lucifer was once a powerful aspect-bearer who rebelled against his lordfather and was ultimately exiled for it, commenting that it is a common enough story among the pantheons.

Additional notes
For additional information on Lucifer that does not pertain to Hades, see Wikipedia's article: Lucifer