My answer pertains only to the modern age Doomsday. I haven't looked into his origins in the New 52 yet:
Doomsday is a bioengineered Kryptonian creature. This is pulled from the superman wiki... I've read the referenced comics before (Superman/Doomsday Hunter/Prey and The Death of Superman) and can confirm the info is correct: You can find Doomsday's bio here
Thousands of years into Krypton's obscure past, long before the
humanoid Kryptonian race managed to gain dominance over the planet,
Doomsday was originally an infant humanoid, probably a Kryptonian; he
was known at this point simply as baby. An alien scientist known as
Bertron saw the environment of his home world. During this time,
Krypton was a violent and harsh world equivalent of hell, where only
the absolute strongest lifeforms could afford to survive. Because of
this, Bertron was influenced to create the perfect creature that could
be suitable for the Kryptonian environment and decided to perform
experiments on a baby. Due to the cruel nature of the experiments, the
baby evolved into a killing monster, with a hatred of all life.
(Superman/Doomsday: Hunter/Prey)
Doomsday was able to escape from Krypton on a random supply ship,
which came regularly to bring supplies to Bertron and his team. From
there, he began to rampage across the universe. (Superman/Doomsday:
Hunter/Prey)
The final planet Doomsday invaded during this era was a highly
advanced planet known as Catalan. All of their advanced technology was
unable to stop him, until they created a superbeing made of pure
energy, who they called The Radiant. This being was able to kill the
seemingly unstoppable monster, and they wrapped him up in funerary
clothes and tied him up as best they could, launching him into space.
(Superman/Doomsday: Hunter/Prey)
Drifting through endless space, the coffin came to land on Earth,
where it remained buried for years, dormant. He spent thousands of years buried,
wiggling his way out little by little until finally escaping and
restarting his rampage (The Death of Superman)
Add On:
Hunter/Prey is where you'll get almost all of Doomsday's backstory. They also did a Doomsday: Year One that builds on it a little more but I haven't read it yet