Blizzard have announced that they will be starting up their closed Beta for Overwatch again, starting February 9th. This time there are few improvements to the experience with a new progression system, two new maps and a new game mode, balancing updates, and more private games - including full AI matches. Players who participated in the first closed Beta will automatically have access to this one as well. Unfortunately, it doesn't seem any new players are going to be able to get in.
While the Beta is closed, the full game releases in late spring. More details on that release here.