You have a couple of options.
Firstly, you could make a series of submitted memes about it, and let people know they're linked by the templates you use, and the titles of the memes.
For example, XenusianSoldier did a series of similar memes about Captain Picard:
imgflip.com/tag/xenusiansoldier%20picard%20series
So, you could make a title like "Grumpy Cat Story" (or something more relevant to the story), and your submitted memes would be like chapters.
Secondly, you could make just 1 meme, and have the whole meme story in the comments.
Here's an example of a "war" between Grumpy Cat and Grumpy Pug on one of reallyitsjohn's memes (you have to read it in between the other comments):
imgflip.com/i/v68oe
If you do this, it's probably better if someone helps you, so you can go back and forth.
Thirdly, you could join the memes together to make one "long" meme, so that it's like a chain of memes, but all together in one place under one submission.
Here's an example of what a "meme chain" looks like:
imgflip.com/i/p0cz1
If you wanted to make a meme chain like that, you make the memes first, then go to www.photojoiner.net to join the photos up.
Maybe have a look at those, and see if one of those is a good format for what you have in mind to do. I'm looking forward to seeing it! :)
