run origin, click the help origin help button at the top menu bar, click on contact us at the bottom of the web page it opens, next, pc/next, Demo/Beta, type in titanfall beta key and just ask for a titanfall beta key to be added to your origin account. you can also chat with EA help and they will do it too, basically they will just hand out a key to anyone who asks, just have origin installed first and a origin account ID so they can attach it for you.
i feel like with 10 hours of played this beta weekend i have seen most of the game I wanna see, bit of a shame. oh well, its a solid game but not really AAA $60, hopefully they start work on Titanfall 2 soon using the Frostbyte engine. That could be a stellar game.