The Apocalypse

A long time ago, humans lived, died, and survived on earth. Humans ruled the world, for a time. Once the nuclear apocalypse struck, everything changed. Humanity was completely obliterated.
The apocalypse did not destroy earth, it merely changed it. Life, somehow, recreated itself and generated new beings to roam the new world, even if it were in shambles. Humans, plants, animals, and even robots turned into horrible monsters. Even some pools of radiation alone were enough to create sentient beings! The apocalypse also created mutations, special abilities obtainable through enough radiation consumption, and portals, a series of mysterious worm-hole-like tunnels that transport anything it consumes from one place to the next. The nature of these portals are unknown, although it may be related to the I.D.P.D., a police force from another world.
No one knows why the apocalypse occurred, but it doesn't matter anymore. As the years went by, the new inhabitants of the wasteland found a new purpose, and some began to travel together. They spoke of an ancient legend: the legend of The Nuclear Throne. The legend begins with a single mutant, who supposedly found the throne, but was never heard from again.
The mutant, deemed "The Proto Mutant", traveled the wastes until he found The Palace. This fortress, heavily guarded by beings of pure radiation, housed the throne. The throne is said to be the only way to stabilize the world once again, and The Proto Mutant had found it! The travelers don't know what happened to the mutant when he sat upon the glorious seat, but the throne did not re-stabilize the world. The travelers concluded that they must reach the throne to save the world, meeting newcomers to travel alongside them. Once a mutant from the group reached the throne, however, things don't go as planned. The Nuclear Throne isn't what it's supposed to be.

Culture

Not much is known from the old world, so the mutated members of this new society of travelers had to develop new ways of communication, and ultimately develop a new culture.
This new language, called Trashtalk, is the language for the new world. At the start of a run, the mutants say "FLÄSHYN" which translates to "Let's do this".
Joonas Turner created the language for this game, and he announced that they are planning on releasing information about its further use some time in the future.

Death In The Wasteland

The travelers know of the risks of attempting to reach the throne, so they have decided to send one member at a time to reduce the chance of failure.
When the travelers send one, or two, of their own to try and reach the throne, it's likely that they will not be coming back alive. This has become a common occurrence in the new world, and the mutants have grown to accept this fact.
Once a mutant of the group dies, they are gone forever. Every time a new run starts, you find the group in a new universe, one where everyone still lives.

The Journey

At some point, the "Proto Mutant" made it to the Nuclear Throne by defeating the bosses of the game, and created the Proto Statues. (We know this from JW saying it in streams, and the tile drawings that occasionally appear in a crown vault, which depict the proto mutant facing the bosses, and the proto mutant itself).
At some point, the Proto Mutant must have died. We can assume this because of the skeleton sitting upon the Throne in world 7-3.
Young mutants are told the legend of the Nuclear Throne, that the throne is somewhere in the wasteland, and that only the best deserve to sit upon it. When the mutants become adults, they begin sitting around the campfire, and one at a time, they start their journey to the Nuclear Throne. All of the playable characters are willing to undertake said journey, and the game itself is what happens during their travels.
Although these mutants firmly believe each portal they enter will bring them closer to the Throne, there's no proof of a Nuclear Throne and it's still unclear if the portals even bring their users to a final destination. The mutants' faith is the only reason they keep looking for the Throne. As the stories go, however, nobody has ever come back from their journey.