Blizzard has announced on the official Overwatch site that the closed beta will be starting on 27 October 2015. The beta will be initially restricted to North America before migrating to Europe and Asia.
According to the site, "In order to get the most meaningful and constructive feedback possible, the total number of players in the Closed Beta will be extremely limited—though we’ll be adding more people from time to time to make sure we have the right number of heroes actively engaged."
Additionally, "These larger-scale Beta Test Weekends will be brought online occasionally for a limited time, generally over a weekend, and will run concurrently with our ongoing Closed Beta test. Because Beta Test Weekends are primarily hardware- and tech-focused, the number of heroes, maps, and gameplay modes will be restricted—but feedback on everything will definitely be welcome."
Gamers can apply for the beta on the official site.