Season 5 of Overwatch will begin on or around June 1st. The news was confirmed on the game's official forums by game director Jeff Kaplan yesterday evening.
In the same post, Kaplan also confirms that the current season will come to an end towards the end of this month.
Blizzard typically likes to leave a week between seasons, so you should expect some downtime between the two, an awful lot of very good players in Quick Play all of a sudden...
Once the date is properly confirmed, we'll update this article. In the meantime, our Season 5 guide contains everything you could possibly want to know about the upcoming new season.