I've been addicted to Starcraft 2 since the Wings of Liberty beta went live back in spring of 2010, so the thought of getting to relive the excitement of new units and game mechanics makes me crazy with anticipation. Unfortunately, only a lucky few will get tp debut the Heart of the Swarm beta to the world.
According to this blog post from Blizzard, pro gamers, members of the press, contest winners or shoutcasters should start checking their inbox. For everyone else, we will have to wait patiently (or not) for Blizzard to determine they need more data and release more beta keys.
Luck will play the biggest role on getting your hands on an invite, but there are a couple necessary steps to take to even be considered. First, you need a Battle.net account, which you should already have if you have played any of the Blizzard games(Starcraft, Warcraft, Diablo). You also need to "opt-in" to beta access by going here. Make sure to run their "opt-in application" that will provide Blizzard with your computer specs. Finally, your chances are increased if you are an avid Starcraft 2 player, so get on and get active.
Blizzard has stated that they will be working with fansites and community personalities to provide more opportunities to join the beta test, meaning that contests may be another way in.
The release of the beta can only mean that the game's release is on the (hopefully) near horizon. If you have to get in on some of the action, try living vicariously through those that have the beta as they stream their games, mostly found here on Twitch.tv.