Before entering this boss room, make sure that you do not have the Piece of Power. While the Piece of Power leads to increased mobility, it lengthens this boss fight because of the recoil it causes.
Slime Eel has four possible holes he can come out of. The hole he chooses is entirely dependent upon RNG. If you select one side of the hole, you have a 50% chance of having the Slime Eel appear on your side. However, his tail swings around the entire room, making it sometimes difficult to maneuver.
Once Slime Eel appears on your side, hookshot his head to drag him out of his hole. Follow this up by hitting him four times with your sword. After eight hits, Slime Eel will be defeated. Ideally you want to be about half a screen away from the hole. If you are too close then you will not be able to be able to get all four hits in.