According to the game information on the Xbox live store here, you will be unable to play the game if you do not have Xbox live. The description for the game on the store reads "Xbox Live Gold required. Currently in Early Access.". This is the same for the Playstation 4 and it is meant to change when the game exists early access. For the moment at least, you will need to have an Xbox Live Gold subscription if you want to play Fortnites Battle Royale game mode.