This is a weird one. The PvE portion of Fortnite (now dubbed Save the World) will be in paid Early Access until sometime next year, but the Battle Royale PvP game mode (which draws a lot of inspiration from PLAYERUNKNOWN’S BATTLEGROUNDS) will be free for everyone when it leaves its current testing phase on September 26th. That’s also when the game will receive squads (currently you can only play solo).
At the moment, Battle Royale is truly free-to-play. There are no microtransactions associated with the game mode at all. In time, they may add some cosmetic purchases, but for the moment, Battle Royale will just be, simply, download and go.
Check out a few frequently asked questions on the official site.