I have a pretty extensive background playing games competitively and I was wondering if this would be something I should include in my resume
It doesn't belong in a resume, really. You WILL be asked about games you've played, if you get an interview. You can mention your background in your cover email - and maybe allude to it in your "profile" section of your resume, if you have one.
That was written before I noticed the "competitively" in what you said. If you were a pro gamer,and you were paid to do it, you MUST include it, as you would any job.
If you did it just for fun and glory, it's worth mentioning, but you'll need to show (by what's in your resume and how you write your cover email) that you are a communicator and technical problem solver, not only a player.
I have no idea how people in HR would view putting things like gaming experience on a resume.
They will probably assume you would get bored quickly and quit.