Lex's motivation and character are so much fun to play with.
For instance, let's say you go with a Lex who doesn't know Clark, and is simply the most powerful bussiness man and mad scientist in Metropolis. Superman's power and persistance in not-dying becomes a thorn in his side, and when he's defeated the first time, he dedicates the rest of his life to destroying Superman.
For another, let's say that Lex does know Clark from Smallville, and part of the reason he can't see through the disguise would be because that would mean a dumb farmboy from Kansas is more powerful than he is. He grows to hate Superman because Superman keeps foiling his plans, but does enjoy the competition for the sheer fun of it.
Personally, I had my own version which was much more about Lex's personality and relationship with members of his family. It's a bit more of a hodge-podge of Smallville, the mid-90's post-crisis Lex, some of Morrison's Luthor and a bit of Cornell's take on the character. Lex and Clark were friends as kids and preteens, but Lex's antagonistic relationship with Lionel Luthor slowly poisoned all his other relationships. They grew apart or became enemies when they became teenagers, with Lex leaving Smallville for a small-scale war on his fairly powerful, but not exceedingly rich father. By the time they meet again as adults, Lex has killed his father and built up his own bussiness empire to eclipse Lionel's memory, and has become the weapons supplier to Metropolis' gangs, mostly so he can also supply the police and politicians with stuff to oppose the gangs. Lex seems to have married early, but after his wife's tragic "plane crash," now raises his young daughter Lena by himself, showcasing his twisted idea of what love is, as he can't understand the idea of unconditional love. He did date Lois for a while, but she's far too good of a person and reporter to be ignorant of his true nature, so she's become an enemy he enjoys opposing on a mental level. He basically treats Clark as a dumb old aquaintance until Clark starts to become close with Lois. His initial antagonism with Superman starts as almost playful opposition, but when he begins to lose the love of Metropolis and his daughter becomes a fan of Superman, he soon decides his beeef with Superman is completely personal.
The actual root of Luthor's sheer loathing of Superman is actually much deeper than that though. Luthor convinced himself of amorality and social darwinism during his battles with his father, and the intervening years have ripened those thoughts into supreme self-absorption and narccisstic confidence in his world view. Because Superman is so blatantly a moral hero, and thus a complete refutation of Lex's worldview, he is compelled to try and break Superman on the fundamental level; somewhere in the back of his mind, Luthor is aware that if Superman is right, than he himself is monstrously evil, and that would terrify him with its implications. Thus, Luthor is constantly convincing himself that Superman is a cancer on existance holding back mankind from seeing its true saviour: himself. Luthor is not insane, and at first he has a firm grasp of most of his flaws with one exception, and that's his pride. Like Smallville and Cornell's Lex, he is also deeply jealous of Clark for his relationship with his parents and his blooming love with Lois.
And the reason why he has a daughter in my version? The Lex Luthor who handed over his daughter to Brainiac in the B13 story-arc was one of the most profoundly loathable slimeballs I've ever read about, and up until then his "love" for his daughter seemed like an almost redeeming quality. And that's the thing- if Superman can ever truly convince Lex that he's wrong, like in All-Star Superman, Lex is capable of actual remorse and heroism. And if the audience can actually tell that Lex is actively choosing to do evil, it makes him that much more powerful of a villain. His daughter is there for that, and we should feel sorrow for the little girl.