To actually get access to the beta, which has already started and continue to add more players, you need to sign up via your Blizzard account management page. Under Games & Subscriptions, scroll down to Beta Access and visit the Beta Profile Settings page. Once there, you’ll see a grid of available betas you can opt into. Once you’ve checked WoW Classic, hit Update Preferences and you’re set.
“To fill our pool of beta and stress test participants, we’ll be choosing dedicated players who meet select criteria from both the WoW Classic beta opt-in and the standard Warcraft beta opt-in. Participants will also need to have an active subscription or active game time on their Battle.net Account. While opting-in to the beta is the primary way to make sure you’re in the running to join the test it doesn’t guarantee an invitation to the closed beta test. We may also consider additional factors such as how long a player has been subscribed to the game so that we have the right mix of players to ensure great feedback toward making WoW Classic the very best experience for the community.”