According to a Reddit submission, EA did send out first closed alpha keys for Battlefield 1. You should read about the Battlefield Insider program if you want to join the upcoming Battlefield 1 beta in late summer, but this won’t help you to get into the Battlefield 1 alpha, because for now they won’t send out more alpha keys according to this Battlefield 1 alpha page. As you can see, the alpha test is exclusive, but some users on Reddit mentioned that they randomly got selected to participate.
It seems to be a very small alpha test phase and it remains unclear if they will send out more keys to expand the phase, but they most likely won’t. But as mentioned, it’s still not too late to get a first impression because there is still the upcoming Battlefield 1 beta in later summer, just check out the Battlefield Insider program to increase your chance to get into the beta!