When Phantom starts the fight he typically floats into the air and then flies to a side of the room. When he is doing this, you want to use some fast movement and rope darts to get to the wall on the same side and quickly hit the ball he creates and throws at you. If you do it right, it should immediately collide and knock down Phantom and allow you to hit him relentlessly.
When Phantom moves to the center and turns around to play his organ, the mask will appear to begin shooting lasers at you continuously for a while. If you perform repetitive rope dart attacks during this phase, you will not be hit by the lasers and will continue to do damage to Phantom.
When Phantom teleports to the top of the screen, he will spawn clones of himself and will rotate like a wheel around the stage. The real Phantom can still be damaged but he will not break out of this attack until at least one full rotation. Once broken, Phantom will sit there for a brief second so you can continue to damage him.