Blizzard is currently gearing up for their first Alpha outside of BlizzCon 2013 for their new MOBA-style "Hero Brawler". Heroes of the Storm will be getting a Technical Alpha soon, according to Blizzard on the game's Official Website.
This Technical Alpha will have a focus on the game's infrastructure and having it run on a "broader spectrum of machines and connection types," so if you do get in, don't expect anything in the build to be final. Alpha is Alpha, after all. According to Blizzard, the game is currently playable, with much of the core experience implemented, but nothing is polished yet.
This Alpha will be limited to the United States for now, with growth as testing demands. Some international press sites and big contributors within the community may also have access, to keep up with development.
In order to have a better chance at getting access to this and other tests, Blizzard recommends that people opt into Heroes of the Storm testing via their Battle.net Account. Blizzard are more likely to send an email with further details of how to get into the Alpha to those who opt in using this method, so be sure to head over to your account and check the opt-in box if you haven't already.