Time to try. I may get some flak for this but that's part of the journey.
Who is He/What Has He Done?
Lucifer is the central antagonist of the dark fantasy TV series Supernatural, which chronicles the trials and tribulations the Winchester brothers Sam and Dean must face to rid the world of evil. When Lucifer comes into the picture, their job gets a Hell of a lot more difficult and trying. As The Devil himself, he was one of the Winchester brothers' greatest adversaries, taking a proverbial backseat only to God and The Darkness (more on the former in another post).
Over the course of eons and 6 seasons, he's done quite a lot of bad things, quite arguably setting the heinous standard for the series, including but most certainly not limited to:
- Corrupting mankind in an attempt to prove to God that they were broken.
- Starting a rebellion in Heaven because God loved humans more than him and went on to blame everyone but himself for the resulting war, his fall, and imprisonment in his Cage.
- Massacring the women and children of Carthage, Missouri, having the men possessed by demons, and ordering them to commit mass suicide to summon Death.
- Violently slaughtering several pagan deities simply out of "disgust" for their lack of loyalty (ironic, yes).
- As a hallucination - although it's also implied it's more than that - he psychologically torments Sam beyond repair, bringing back the latter's memories of being stuck inside the cage with Lucifer and benign tortured for hundreds of years in (in Hell time).
- After Castiel takes on Sam's broken psyche, "Halluicfer" (I know, I suck) then mentally tortures him similarly.
- Soon after escaping his Cage once more (via possessing Castiel), he keeps Crowley as his "dog", abusing and demeaning him.
- Attempting to kill Sam painfully by siphoning his soul power and lying to Dean upon his return that it wasn't intentional.
- Possessing and posing as rockstar Vince Vincente and performing various disturbing activities, such as having a fangirl of Vince carve his name into her chest and playing "quit stabbing yourself" with Vince's band's producer.
- Possessing President of the US Jefferson Rooney and conceiving a child with Rooney's secretary (who, of course, thinks he's Rooney). Said child is one of the most powerful and potentially dangerous beings in the universe.
- Showing his true form to Rowena MacLeod, terrifying her before violently killing her, stomping her head in "until the white meat showed" and setting her on fire to ensure she stayed dead (she did not).
- Lies to the Host of Heaven that he can create more angels and restore the angels' wings if he becomes the ruler of Heaven.
- Stealing Jack's nephilim grace after Jack rejects Lucifer as his father, then flying away with him and Sam to an abandoned church where he assaults them both and demands the two fight to the last one standing.
- Planned to destroy the universe and remake it in his own image; if ever he wasn't, by now he's gone full-blown Chaotic Evil and is no longer inhibited by regard for anybody's life, least of all his own son's.
Why He Doesn't Qualify
He wants his father to love him. And no, I don't mean that as in "he's a bratty child who wants all the attention on him” (albeit, he is), I mean that as in "he's literally cried more than once at God 'abandoning' him". He had even attempted to save His life from Amara/the Darkness after forgiving Him (briefly) for his condemnation. Back to the crying: once in season 12 when possessing Vince Vincente, after God leaves Earth to reconcile with Amara, he admits he has no true plan, except to just "keep smashing [His] already broken toys", all while barely holding back tears and shouting about how He "ditched" him. Another instance is in season 13, wherein his archangel brother Gabriel, fed up with Lucifer trying to make himself seem the victim, calls him on his obvious lies, stating he couldn't stand that God loved humans more than himself and compares him to cancer cells that need to be cut out to stop their spreading, but that it was too late then and it's too late for Lucifer to change his ways now. Lucifer is evidently keeping the waterworks at bay but visibly sheds a tear afterwards.
In season 15, God revives Lucifer (he was killed in the season 13 finale) and delegates him the task of retrieving His death book from the Winchesters, which Lucifer carries out, proclaiming himself "the new favourite" to Michael and calling him a cuck like a 13-year-old sibling provoking his older brother over nepotism, which he essentially is, after getting the book into his hands. So, seems he's still at least a bit hung up on acceptance from padre.
I could also go into some depth about the seeming love he still held for Gabriel and even reluctance to "kill" him (one of Gabriel's tricks), but by the time Gabriel's final hours are in bound, their brotherhood is lukewarm at best, and even when Gabriel flames him into shedding "I'm a tough guy" tears, what upset him more was Gabe's accurate description of Luci's relationship to their dad.