There are multiple ways to get into multiplayer Beta of Gears of War 4. The game's beta test is divided into 3 testing periods in which you can participate. As Coalition states in the official forum post, the code distribution is starting on April 18th.
The codes from the first wave of the Beta test (from April 18th to April 24th) will be sent to the players that have played Gears of War: Ultimate Edition. The developers say that they plan to send out keys in the chronological order. This is done to make sure that the servers will hold up the influx of players. The codes will be delivered via Xbox Live Messages.
The next wave of the Beta test will start on April 25th and will last until May 1. All Xbox Live Gold players can play the game during this period of time.
Lastly, Coalition scheduled a live stream for today (April 15th). During the event, the studio will hand out keys that will allow users to play the Beta three days before the first period begins.