According to statements listed on the GameStop Germany page for Call of Duty: Black Ops 4, the beta will begin in August. In addition, the PC version will have an open beta.

The site says that the launch of the private beta for the PlayStation 4 and Xbox One versions will take place in August. It goes on to say that the PC release will not need beta codes because the PC beta in particular will be "open" for all Battle.net players.
Availability and start date (s) of the beta on the respective platforms may change. For more details visit www.callofduty.com/beta. Minimum duration of the beta: 3 days. Limited time, as long as there are beta codes. Internet connection required.
For the PC version, no codes are sent because no code is needed for playing.

Guide to BETA participation:www.callofduty.com/beta

The exact dates of the beta are not yet known.