Grim Creeper sends out six bats in different formations at you. He will not flee the battle unless you defeat all six bats. The easiest way to do this is to position Link in the top-right corner of the room near the door, face downwards, and swing wildly. After beating the boss, make sure not to jump through the topmost door that opens, otherwise you will glitch out of dungeon bounds.
What is a normally difficult battle is made a lot easier if you have the Magic Rod, the main item of Turtle Rock. When the dialogue before the fight finishes, go down to exit the screen and return back up to the boss room immediately after. This will skip some animations and save time. The strategy after this is dependent on if you are doing a No S+Q category or not.
Stay on the ladder and not the top platform. Spam the Magic Rod to the left and right alternatively in order to hit the boss before he reaches you. Three hits will defeat him.
Similar to the above, but this only works if you have performed a Save and Quit after opening the dungeon. Strike the Evil Eagle when he is directly above you. As he flies upward, shoot down with the Magic Rod in order to hit him when he is off screen. Three hits will defeat him. This strategy also works with the Hookshot, but it takes six hits total instead of three.