There are a few things that must be part of your post-apocalyptic story, or you have no story. Let's take a peek.
#1. An apocalyptic event. That's right, you can't have a post-apocalyptic world without something that got them there. What will yours be? Viral, bacterial, natural, man-made, space-related or nuclear? These are all options, and there are probably plenty more. Did the swine flu get out of hand? Was it helped by humanity or just one of those things that happens in nature? Did the Earth tilt too far off its axis? Did nuclear Hell flame rain down upon the continents? There must be a reason the people in your story are stuck in this particular landscape.
#2. A time frame. Are they living through the event or has it already happened? Is it fresh or decades down the line? You have to know when it happened and what stage humanity is in to really tell your story. If it happened decades ago, the landscape is going to be significantly different than if it just happened yesterday. Quality of life will also probably be very different. If they've been coping for decades, they probably aren't struggling to find food or water sources as much as if it just happened and everything is tainted or burning. If it's a new problem, there will be mostly individuals and small groups, whereas a length of time may mean there are established towns/cities.
#4. Strong characters. We need to believe that these people can make it (or not, as the case may be). It must be a real struggle. We have to care whether they can survive, one way or another. Maybe we hate this guy so much that we question why he survived, when better people died. Maybe we love this character and desperately want to see her rebuild her life. Whichever characters you have, we must believe in them, and they must have a mission, of sorts. Does Evil Guy want to take over what remains of the world? Find natural resources to survive? Or just be left alone? Does Lovely Heroine have a child to fend for? Is she just trying to find a home she can call her own? What drives them? What are they trying to accomplish? This is important in every single kind of story you may write, but don't get so intent on your world building that you forget your characters.
#5. A purpose. Alright, we get it. The world has ended. The apocalypse has found us. Whoopty-doo. What is so important about this world that you just have to tell the story? What are we going to take away from this? I'm not talking about a moral (necessarily), but just a life story that means something to us when we read it. A violent post-apocalyptic world, where survivors are constantly under siege, does us no good if we don't come out of the story feeling something. Perhaps you want us to know that humanity will always find a way to thrive. Or that love will always pull someone through. Whatever it is, make it part of your story.
this link should take you to a comprehensive list of classic post-apocalyptic stories. Of course, The Hunger Games and Forest of Hands and Teeth should be on there. Also, I read Without Warning by John Birmingham recently, on a whim, and I enjoyed it. It was more a political/government/military-type book that took on what happened in those facets, so different than I'm used to for this genre, but also quite good. I don't know how The Marbury Lens and The Maze Runner are qualified, but I'd consider both to be sort of post-apocalyptic. We really aren't sure with The Maze Runner, but we get a sense something big must have happened, and in The Marbury Lens, the alternative world he visits via the lens seems quite post-apocalyptic. Both are excellent books, though be aware that The Marbury Lens can be graphic or disturbing, despite being Young Adult.
The short of it is, fully realize your story so we can be drawn into it, feel for your characters, smell the fires, feel a sniffle coming on as everyone dies of the Hulk of flu bugs. Watch some of these movies or read some of the books (or both) and figure out what you like in them, so you can duplicate that, in a sense.
In your opinion, what are other important aspects of a post-apocalyptic tale? What books or movies might you recommend?
May you find your Muse.
* Four Horsemen of Apocalypse, by Viktor Vasnetsov. Painted in 1887; Viktor Vasnetsov [Public domain], via Wikimedia Commons
** Stalingrad after the battle; See page for author [Public domain], via Wikimedia Commons
***The aftermath of Hurrican Camille. Ruins of Texaco gas station with Rambler automobile, Biloxi, Mississippi, 17 August 1969