Warning: spoilers for Batman v Superman: Dawn of Justice are ahead!
Just like other versions of Lex Luthor in the comics and other media, Jesse Eisenberg’s version had a basic goal in Batman v Superman: Dawn of Justice: to embarrass Superman and eventually see him dead. When pitting Batman against the Man of Steel didn’t work, the LexCorp boss unleashed Doomsday on the two heroes and Wonder Woman. Compared to his other plans in the movie, creating the Kryptonian monstrosity didn’t make as much sense, but Eisenberg believes Luthor’s obsession with planning ahead was the main instigator.
Although he admitted that his Lex Luthor became slightly crazier as Batman v Superman: Dawn of Justice progressed, Eisenberg also explained to IGN that bringing Doomsday out was Luthor’s final, desperate way to eliminate his alien nemesis. He said:
I think Lex becomes increasingly unhinged throughout the movie. I also think he’s a guy who has 40 back-up plans and so when one thing doesn’t work out he has another and if that doesn’t work out he has another, which is why I think he never feels that threatened by Superman and Batman because he knows he always has the leverage and his final act, in my opinion; and this is now thinking back a year-and-a-half – was this kind of like last-ditch effort to leave it all on the table.
As a refresher, Lex Luthor created Doomsday in the abandoned Kryptonian ship’s Genesis Chamber using General Zod’s corpse and his own blood, resulting in a "Kryptonian Deformity." As a killing machine meant to be sicced on Superman, Doomsday’s existence certainly makes sense. However, what would have happened if Batman had killed Superman as Luthor wanted? Doomsday was nearly finished forming by the time Superman flew back to LexCorp, so if the Man of Steel was dead, then Doomsday would still have rampaged towards Metropolis, and there was no way that Luthor could have contained him. Maybe Wonder Woman and Batman could have defeated him since they still had the Kryptonite spear, but it still wouldn’t have been a guarantee. Now, if Luthor had begun creating Doomsday after the Batman/Superman battle, then that would have worked more efficiently as a "last-ditch" contingency, but by that point he would have been arrested. Perhaps Luthor should have just used Doomsday from the start.
While the "what-if" scenarios are fun to ponder, the fact of the matter is Doomsday was defeated in Batman v Superman: Dawn of Justice, but only because of Superman’ sacrifice. As for Luthor, Eisenberg was right about him being "unhinged." Throughout the movie, Lex’s eccentricities were on full display, and by the time he was thrown in prison, he seemed to have gone fully off the deep end. His babbling to Batman sounded like the rantings of a madman, though when considered with Batman’s "Knightmare," we know this is hinting at Darkseid coming to Earth.
Although Lex Luthor is still alive, Jesse Eisenberg recently said he’s not sure if he’ll return for Justice League: Part One, so we’ll have to keep waiting for news regarding another appearance. As for Doomsday, the monster seems to be dead, but if he’s like his comic book counterpart, we may not have seen the last of him.