The battle starts with Byrne jumping along pillars and occasionally shooting weak beams towards Link. Just dodge these and eventually Byrne will attempt to hit Link with his massive gauntlet arm. Just dodge out of the way and the gauntlet arm will get stuck in the ground. Use the phantom and run over to the gauntlet. Zelda will pull on the gauntlet cause Byrne to fall to the ground, so run on over and get a few sword slashes in before he awakes. Repeat this process until Byrne stays on the ground level.
Just run up to Byrne and slash at him repeatedly. Hell block your first two attacks but the third one will usually hit. After Byrne is a bit shaken up, he will now perform large purple charged attack towards Link. When he is charging up, go hide behind the phantom. When Byrne charges in, the phantom will grab him and hold him steady. Run on over and deliver some sword slashes while he is stuck.
Afterwards Byrne will get a bit quicker with his attacks and does a better job of blocking Links sword attacks. Just keep at it and keep slashing away. Byrne will eventually be defeated without much of a sweat.