All you need is a good knowledge of the program, and some drawing talent. And actually being able to carry through with the project. As you draw everything to make your Flash animation, you need to draw, well, a LOT of things. So it can be long.
However, when you pay tribute to a video game series (especially the 2D games), you have a big advantage. There are numerous websites that make it their mission to collect sprite sheets. What's a sprite sheet? It's basically a picture on which you have every possible sprite that can appear in the game for a character, for the enemies, for the fields, for the backgrounds... In theory, using Flash and all those sprites, you could basically re-create the whole game from scratch.
But that's not what they use it for. No, it's better than that: They make Flash movies out of those sprites. I particularly like sprite movies because, just like every Flash animation out there, they are basically like animated fanfiction: The story is written, the sprites are taken from the games, the animation is created, and so on... That's a particularly impressive task to do such a thing, because, well... the amount of work is enormous. That's probably why series such as Super Mario Bros. Z took so long to be completed: There was almost only one man behind all this awesomeness. A man nicknamed Alvin-Earthworm. That's also why it has gained such a cult following: It's just such a huge love letter to both the Mario series, the Sonic series and the Dragon Ball series.
He didn't actually finish the whole thing, though. However, he still made over two hours and a half of Flash animation, which remains pretty darn impressive.
Of course, not all of those Flash films are the same quality. Just like fanfiction, it can greatly vary between fics. Some Flash animations are definitely... more childish, or more violent. Or both.
If you put aside the unnecessary gore and toilet humor in Rise of the Mushroom kingdom, you still get something pretty funny and creative.
Anyhow, keep looking for Flash animations with the characters of 2D videogames that you like. You could be very surprised of what you will find.