The reason Lex Luthor wants to kill Superman will always be the same. He does not trust an alien to protect human kind. (Xenophobic much?) He thinks humans should be able to do that themselves and he wants to be the one to do it.
Hope this answer from Quora clarifies even more this part as he elaborates it a lot. Although this very strongly related to the comics, a huge part of it also works for the movie.
Now, for why he created Doomsday... Well, with some research, I found out that not even Jesse Eisenberg is really sure why this happened, although he thinks it made sense for the character.
In his own words:
"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." - Jesse Eisenberg, to IGN as seen in Cinema Blend
So according to this, basically what he believes is that Lex Luthor as a master planner, probably had a contingency backup plan behind Doomsday's creation.
It could be true, since he created him from his own blood. Maybe it just wasn't clarified in the movie.