I haven't played the original Alan Wake, but as the American Nightmare is on sale now, I thought of buying it. But the question is, how much do I miss of the story if I only play American Nightmare without playing the original?

Pretty much (although, I didn't think that this has so much story at all) but it still is good standalone game. Instead playing this, you should pick normal Wake. Which is great.

Edit: mostly about action this one.

You would miss quite a bit. At the very least, the entire plot of the original would be spoiled for you.

I really recommend playing the first game first. Might not hurt to pick this one up in a sale now, but I would hold on to it until you've caught up.

The whole AW thing is mainly story, it's based on a novelist too, but its not really scary, just very heavy on atmosphere and story, you need to start at the beginning.

I found the original Alan Wake to be awesome, but American Nightmare felt like a silly, not-scary nightmare that didn't make any sense in the few hours I played before stopping out of "this is dumb, I have better things to do". I'm guessing the way the game is silly was intentional (even the name suggests it's a dream), but I didn't like it.
Anyway, I'd say that if the beginning of American Nightmare has a story, then a department store shopping list does as well.

/edit: to clarify, if you're interested in the story, don't start with American Nightmare.

Nightmare is like it would be some DLC but a good one.

Perhaps I get it to my backlog then. Thanks.