Alpha invites have apparently already started to go out via Email. Apparently at this point, the only way to maybe get in the Alpha test is to subscribe to the new letter and make sure that your Veteran status is up to date; only then maybe you might be called upon. DICE will not be giving out any public keys to the Alpha test, so this is the only way that you might get in if you have not already been invited.
As for the Beta, it will start in September, and those that purchase the Limited Edition of Medal of Honor will be able to join the Beta once it starts. Rumors of an Email going out about the Beta are apparently true; however, there was no code or ticket in this Email that will get you into the Beta. DICE will be making an additional announcement about joining the Beta as we get closer to September.