As a former game writer, it makes me sad that nobody has answered this question yet.
There is all kinds of writing that goes into making a game. There is the technical side, which includes things like instruction manuals, doing copywriting for ads, blogs to promote the game, and all sorts of other things. On the creative side, you have scripts for games and lines of dialogue for the characters.
Whichever you prefer to do depends on the kind of writer you are. I enjoy anything to do with marketing and creativity, so I did both.
The way you get into a writing position in a gaming company is to just apply. However, there is a lot more to it than that. If you want to write for most gaming companies, you'd better have some experience under your belt. For big game companies, you'd better have a lot.
One good way to get started is to just gain some experience. Pay attention in your high school english classes, or if you're in college, take some creative writing classes. Try to learn more about writing in general. There are many good books on writing, especially game writing.
One thing you want to do is build a portfolio. There are lots of ways to do this. The best ways are to start blogging about video gaming or a topic you enjoy, write fiction, write screenplays, write just about anything to show you are dedicated to you writing.
Not only this, but you should explore other writing avenues as well. For me, I was able to get my foot in the door because of the experience I had as an ad copywriter. The more diversified you are, the better your chances of landing a game writing job.
If you can do it, it wouldn't hurt to pick up some technical skills as well. You don't have to do hard programming, but learning some basic scripting languages will help you. I don't know what's popular now but when I was in it Lua was a popular learning language.
Because of the direction games are going, game writing is possibly a dying profession. It's a shame, because I found it really enjoyable. But I found the jobs I got because I looked for them. That's what you have to do. When I first started, I just typed in "game writing" into google and searched all the pages. It's how I landed my first job.
I don't do it anymore, but when I did, I found the profession very rewarding. Even though I didn't make a lot of money from it (If you want to know what I do now I self publish fiction).
If you have any questions let me know, I could possibly help you out.