Ah, girl, I got you covered. I've had to take care of a few of these people in my time.
1.) Be Savage
And by Savage I mean low-key rude. Since their your frenemy, you should know a little bit about them so use that to your advantage. Are they short? Great! Are they tall? Great! Did they embarrass themselves in class several years ago? Even better! Bring it up!
2.) Don't be a two face back stabbing frenemy back
This one is important. If you don't let on that this person is getting on your nerves, it makes them mad and that makes it even sweeter. Now, I know this probably isn't what you wanted to hear but just try it. Ignore them so hard they start to doubt their own existance.