Keep the sound on. This is your friend, and it will help you a lot. There are three ways to get past the machinegun fireballs, there is the easy way, which is standing in the bottom left corner, the medium way, which is standing on the left side and after he finishes talking run to the right, up the vine and down the left, and the hard way, which is climbing the vine and standing on the platform at the top. (At least, I found it to be the hardest of the three.) After he is done the machine gun blast (It will go just over 3/4 of the way around the screen) stand on THE VERY BOTTOM FLOOR and wait for him to set the platforms on the left and right on fire. Jump on the pillar and fire. Eventually, you'll learn how to time it properly to dodge the fireballs while still hitting the eye.
The fastest way to shoot out the eyes is to jump up, shoot three or four times (Depending on how fast you can mash) wait to fall a tiny bit, do the double jump and shoot four or five more. Three of these should take out one eye.
