Overwatch Beta Starting Later this Month
"The Closed Beta will be composed of a small number of testers who will have regular access to our public beta test" according to Blizzard, and only be allowing for an "extremely limited" number of players to get in. After the opening of the beta on the 27th, Blizzard plans to slowly increase the number of players while making preparations for Overmatch's European and Asian release.
If you don't get in initially, don't worry too much. On specific weekends, Blizzard will "open the floodgates" as they put it, and allow for anyone and everyone to hop onto the servers to test their hardware and see how things are working.
The Overwatch beta will be playable on Windows only, and "the number of heroes, maps, and gameplay modes will be restricted," but Blizzard is looking for as much feedback on what they give you as possible, so you're being encouraged to join if you can.
You can sign up for the beta here.