In the beginning, everything went as in canon; Lilith 'disobeying' Adam and fleeing the Garden. However, Lucifer didn't go to comfort her, seeing her independence as arrogance rather than a personal strength. Instead, Lucifer admired Adam's stern and dominant nature, approaching him and trying to temp him into choosing him over Eve - he's an angel, so can also reproduce and 'start humanity' with him too. Adam rejects him, being the world's first 'no homo' bro. Out of spite, Lucifer gave Eve the Apple, hoping that she'd be seen as a troublemaker and be kicked out of Eden, leaving Adam alone and with no other choice. It didn't work, as both himself and Adam were also banished - Lucifer of course being banished from Heaven entirely and sent down to Hell.
Hell was a barren landscape of rock and dirt, leaving a very bitter and jaded Lucifer to create his own inhabitable world. It took a few human generations before Sinners began appearing, and at this point Lucifer was extremely scornful of any descendants of Adam's coming into his realm. But, like in canon, he cannot harm Sinners.
In the only retaliation he could do, Lucifer corners off his own 'Eden', leaving Sinners to roam free for centuries. He had his space, and Sinners could make do with whatever rocks they could bang together for entertainment out in the wastelands. In his own 'Eden', Lucifer created small copies of what he could glimpse up to on Earth, eventually manifesting his own palace and small eco-systems of forests, deserts, mountains, etc.
It was only after a few more hundred years that Lucifer began to notice Sinners becoming more prevalent and nosey - the occasional Sinner wandering into his territory out of desperation for any kind of resources outside of brimstone and fire. Initially Lucifer would scare them off, but after a few more decades, he decided to have some fun.
The occasional intruder would be given an offer: resources for their undying loyalty.
He would be their king. Their God.
Of course the Sinners took him up on that offer, taking resources back to their 'homes' and spreading the word of Lucifer. Eventually, it became widespread that this Fallen Angel, this King of Hell, would take care of them for a price. Offerings began being left at the boundaries of Lucifer's realm, and with each one the beginnings of a more survivable Hell started to emerge. Rivers of boiling water began to carve through rock, a hazy red replica of the sun appeared in the otherwise barren sky, foliage fought its way through the ground in small green shoots, a stark contrast to the insistent red.
However, along with these offerings⦠appeared six creatures that Lucifer knew weren't Sinners, but didn't remember creating neither. They were small and kept to themselves, living within the palace. At first Lucifer tried to get rid of them, but it was a fruitless endeavour, eventually giving up and reluctantly becoming fond of the critters, who in return stopped pranking and annoying him (as much). These are the other Sins - Satan, Asmodeus, Beelzebub, Mammon, Leviathan and Belphegor.
As humanity evolved, and the amount of Sinners began to grow, so did Hell. Villages became towns, which then became cities. Sinners were appearing far faster than Lucifer could create resources for, and the concept of Gods and Kings were far less appealing to the general population which had shifted from petty thieves and liars to rapists and murderers.
Offerings stopped, and so did Lucifer's tolerance for these parasites which had infested his home, cities which had overtaken the grandness of his Palace and technology which had made his own powers of creation less awe inspiring. Overlords began to crop up, taking other Sinner's souls for themselves and developing powers similar to Lucifer's own⦠He was furious.
Overwhelmed with anger, hurt and bitterness, Lucifer demanded an audience with Heaven. With Adam. If he couldn't kill Sinners, he knew the First Man could.
Lucifer proposed the idea of the Exterminations, as he wanted Sinners to remember their place in his kingdom. Adam accepted, because of course he did, and the yearly Exterminations began some time during the 1900's.
It was during one of these Exterminations that a freshly created Exorcist mistook Lucifer for a Sinner, managing to land a very minor hit. It didn't cause any long-term harm, and the Exorcist was promptly dealt with, but it had Lucifer thinking⦠He wanted an heir. Someone who wouldn't leave him, someone who would need him forever and always. Someone who, if the worst were to happen, would continue his legacy. His bloodline. Something he was denied back in Eden.
Disguising himself as a Sinner, Lucifer found a random citizen who took his fancy and lay with him, conceiving his heir⦠and promptly strung the Sinner up on the tallest tower in Hell on the next Extermination Day, ensuring their demise. He didn't want the complications of the sire figuring things out and demanding marriage or anything, after all. Charlie was born, and Lucifer declared to all of Hell that he - the only being in Hell who could create new life - had secured Hell's future with their new Princess⦠but Hell didn't care. So, he took his daughter and raised her privately.
Initially Charlie was raised to be just like her father; cold, uncaring, and to hate Sinners⦠but Charlie wasn't like that. She smiled and brought her father flowers which she picked with the help of the mini Sins. She laughed and ventured outside the Palace to play with Sinner children (what kind of God allows kids to go to Hell, Lucifer always thought). Charlie was everything Lucifer once was, but had lost a long time ago. How could he curse her to the constant anger and resentment he felt? So, Lucifer let Charlie grow up how she wanted, keeping her safe and happy, until she too grew up and wanted to start her own life away from the Palace.
Lucifer was reluctant, but he couldn't deny his little girl anything, so let her go⦠Only for her to open a hotel, inviting him to stay! Charlie knew her father wasn't happy, and wasn't sure if he ever had been, so decided that if there was anyone she wanted to 'redeem' it was her beloved father. Of course, she didn't tell him this, leaving out that tiny little detail to instead focus on spending time together out in 'the real world'.
Lucifer hates 'the real world', a world where he's not feared nor worshiped. Where his status was scoffed at and his powers are dime a dozen. But, he'd deal with it for Charlie. And so, the Hazbin Hotel was started, with a very grouchy, entitled King staying up in his apple tower the majority of the time.