Blizzard announces the Overwatch beta will begin on October 27th in the Americas, with testing the teamplay shooter in other regions to follow. They say they will recruit a small number of core testers but that they will also be opening the game up to more players for some stress test weekends along the way. This page has all the details along with instructions on signing up if you are interested. They say:

We're aiming to accomplish two primary goals with our public beta test: First, get tons of top-notch feedback on the gameplay—including balance, feel, and fun—to help us make Overwatch the best team-based shooter on the planet. Second, we want to hammer the heck out of our tech, including stress-testing our server infrastructure and making sure the game runs great on the widest variety of systems possible.

To accomplish this, we're dividing our beta participants into two different groups, each with different mission objectives and deployment schedules. The Closed Beta group will form the core of our testing crew. In addition, we'll be conducting a number of Beta Test Weekends with wider groups of players when it's time to break out the big guns.