Post-apocalyptic settings are more commonly associated with science fiction than fantasy. Whether nuclear holocaust, biological warfare, or some other horror, we keep imagining ways we will destroy ourselves in the future. Nevertheless, there is no reason why your Dungeons & Dragons campaign can’t use the trappings of a post-apocalyptic world as well.
The first thing you need to decide when creating a post-apocalyptic campaign is what the world was like prior to the devastation, and perhaps just as importantly, what caused it to be destroyed. There are infinite possibilities for what the world used to be like, but the most common choices are fantastical or technological.
Starting with a standard fantasy world pre-apocalypse will probably save you a lot of work. The cause for the devastation should be equally fantastic. Perhaps the world was collateral in a war between the gods. Maybe creatures from the far realm broke through and have claimed vast swaths of the changed and twisted landscape. Perhaps, like Dark Sun, sorcerer kings drained the land of its life.
Another possibility is a formerly technological world which has taken on fantasy elements after an apocalypse. Magic may be real, it may be misunderstood super-science, or a mixture of the two. The cause of the apocalypse may be anything from the standard nuclear holocaust to something more unusual, like the development of portal technology breaching the walls between dimensions and allowing all manner of horrors in.
Using post-technological fantasy worlds in D&D does require a lot more work on the part of the DM though. The DM will need to figure out how technology works and how it interacts with magic. Is use of technology forbidden due to its association with the holocaust? Is it just another power source, right next to martial, divine, and arcane? Is all “magic” simply misunderstood science? The DM needs to answer all of these questions prior to the beginning of the game.
After determining what the world was like, the next thing to decide is how badly the world is damaged. The worse off the world is, the further from standard D&D it is, and thus more work has to be to put into it. However, the more extreme worlds tend to be the most memorable as well. I tend to split my post-apocalyptic settings into three broad categories: Ruined Earth., Extreme Points of Light, and Recovered Earth.
In a ruined earth setting, civilization is still decimated by whatever destroyed the world. While some outposts of “civilization” may exist, these are usually brutal enclaves, like those found in the Dark Sun setting or the Mad Max movies. Normally, the strong rule the weak, and no true justice can be found. Often, some form of gladiatorial entertainment is the closest thing that passes for the courts. In a ruined earth setting, the characters probably have banded together for survival, rather than a desire to acquire riches.
As you may already be aware, the so-called “Points of Light” setting assumes a dangerous world with occasional bastions of civilization. The “Extreme Points of Light” assumes a world in even more dire straights, but with some form of civilized society still in existence. Usually, there is some insulating factor that kept civilization extant while the rest of the world burned.
One example I can think of is the comic book Meridian. In that world, the surface of the planet had been devastated, but civilizations still exists far above the surface on floating islands. The islands were interconnected by a fleet of airships, and few people born in the clouds ever traveled to the dangerous surface below. Meanwhile, life on the surface is much like the ruined earth setting described above.
The closest to the traditional D&D campaign is the recovered earth. In this setting, some time has passed since the cataclysm and the world has had some time to recover. Despite this recovery, some areas of the world are still marked and some creatures still changed by the events. Sword & Sorceries Scarred Lands is a good example of this setting. Even the post-Spellplague Forgotten Realms has some aspects of this, with unpredictable Changelands and creatures sporting Spellscars.
Any of these settings can offer a change of pace from your typical D&D campaign. Hopefully, the tips above will help, but if you need additional inspiration there are several pre-made settings with apocalyptic themes to check out. Dark Sun (AD&D 2E) and Desolation (Ubiquity) are definitely good places to start.
Or I suppose you could just watch a Thundarr the Barbarian marathon on Boomerang.