Register on auth at https://origindot.space using your main EVE character.
Once you’ve done this, on the left hand side you’ll see a services button. Click this and proceed to the forums using the link provided. If you’re not logged into the forums automatically, click Log In in the top right. You should be automatically signed in with your main EVE character.
You’ll now have various options to different corps recruiting within the alliance. Follow their individual directions to apply to the corporation you’re interested in!