Respawn Entertainment have updated the Titanfall website to allow you to be part of the process. They have also announced that they are not in the Beta phase, but that this is simply a small test for the public.
When you sign up you will receive an message stating: "Your application to participate is complete. If you are selected to take part in our upcoming test, you will receive an email in the coming days with a unique code and instructions on how to download and access the game on Xbox One. Thank you!"
You can sign up on their website, here. Titanfall is out in March 2014, but a date for the alpha trial has yet to be announced.