During this battle, the Bilocyte’s primary motive of dealing damage will be to shoot green orbs of poison at you. Begin by moving to around the middle of the platform. There, you are to counter the orbs by swinging at them with your sword. Hit one with a left-right horizontal slice, and another with a right-left horizontal slice, to hit the boss’s head wings. Then hit another orb vertically to stun the boss, then go up to it and bash it with your sword.
Repeat the above process and deal enough damage to the eye to continue.
For the second phase of the battle, move as far away as possible from the boss. It will continue to throw the yellow orbs at you, but also will move after shooting the orb. Using the same “physics” as before, hit the orb at the head. You can continue to do this for a while. Alternatively, since your Master Sword has a long range with the Skyward Strike, you can attack it directly with those. However, that would put you in more danger, but it WOULD end the battle quicker.