He eventually crashed on Earth and the rest is history. In Smallville, General Zod and Faora, from a combination of monsters found on Krypton, turned Davis Bloome into Doomsday. So if you combine both stories then General Zod becoming Doomsday through Lex Luthor's experimentation makes sense.

Does General Zod become Doomsday?

In Batman v Superman, Lex Luthor creates Doomsday using his own DNA, that of the dead General Zod (an alien from Krypton you may remember as the bad guy from Man of Steel) and Kryptonian technology. ... He's nicknamed Doomsday by the media after he comes to earth.

Who sent Doomsday to Earth?

It has been hinted at that General Lane sent Doomsday after the Kandorians in the first place, and the creature is only one of the "weapons" at Lane's disposal. General Lane put Lex Luthor to work on apparently "improving" Doomsday who, by the end of New Krypton, had still not awoken from his most recent death.