Lucifer Morningstar

Lucifer is the ruler of Hell, the husband of Lilith and the father of Charlie.

Background
Aside from his position as Hell's ruler and being the husband and father of Lilith and Charlie, little is known of Lucifer. The show's creative team, however, has described him as "goofy" and similar to Willy Wonka in ways. He is also said to have a great love for music, being a talented musician himself.

Though he plays no role in the pilot, Lucifer is referenced and alluded to many times. Shadowy figures resembling him appear during the opening song "I'm Always Chasing Rainbows". Notably, during the verse "Why have I always been a failure", his shadow materializes behind Charlie, possibly symbolizing disappointment in his daughter. Later during the fireworks display, four demons are seen watching from a window, one of which is a red-eyed silhouetted figure wearing a large hat and holding a distinctive cane, features that strongly suggest it is Lucifer.

His visage is later glimpsed in several background images of the Happy Hotel, most notably a family portrait that catches Alastor's attention.

Charlie
Charlie is Lucifer's daughter. In the pilot, it is implied that Lucifer does not believe in his daughter's plan to rehabilitate sinners. Nonetheless, Charlie did take his words of dealing with unscrupulous folk to heart, such as Alastor.

Lilith
Lilith is Lucifer's wife. It's unknown what their relationship is outside of being married.

Alastor
Lucifer has yet to physically interact with Alastor. According to Vivziepop, Alastor would kind of like him. In one of Viv's streams, Viv mentioned Alastor would like Willy Wonka because he's a jerk, and then she said that Lucifer has a lot in common with Willy.

More than that, Vivziepop and Dave said that Alastor in no way would be afraid of Lucifer, though Lucifer may be an intimidating character, there's mutual respect going on between them.

Trivia

 * The apple tucked in the brim of his hat and on the end of his cane seem to reference the story of Adam and Eve, where a talking snake (often interpreted as being the Devil) persuades the first woman Eve into eating a forbidden fruit, which is often depicted in pop-culture as an apple, resulting in her and Adam being thrown out of paradise.
 * Lucifer's character is based on the pop-culture version of the Devil, where he is typically portrayed as being the fallen-archangel Lucifer who attempted to conquer heaven in a great war before being cast down to Hell where he became its ruler and the adversary of God. In some versions and renditions, Lucifer is dubbed "Satan."
 * In Hazbin, Lucifer and Satan are separate characters.
 * For full Biblical clarification, Lucifer (Shining One) is only used once throughout the entire Bible in Isaiah 14, where Verse 4 makes it clear that it refers to the King of Babylon, who held similar ambitions to Satan, with Verses 16 to 20 reinforcing that the term refers to a Human King. Thus making Hazbin's depiction of Lucifer and Satan as separate entities Biblically accurate, even if only by accident.
 * In her VIVZIE STREEM- Test Test -#1 live stream, when asked if she can tell fans anything about Lucifer, or at least what his relationship is like with Charlie, Vivziepop said that she can not say very much about Lucifer yet because of it being a little bit too much of a spoiler for the future (especially his relationship with Charlie, which is kind of a big part of the show). The only hint Vivziepop gives regarding Lucifer's personality is that he is goofy and enjoys polka music.
 * Charlie seems to have inherited his rosy cheeks, yellow eyes, fangs, and pale complexion.
 * He is slightly shorter than Charlie.
 * Lucifer can play many types of instruments very well, but according to Vivziepop, there is an instrument he has not perfected yet and is quite bad at it as a result. Additionally, she said that his favorite instrument depends on who the voice actor is.