The Moon Laser game is found in Nene's Fortress, at the end of disc 2. It is basically the boss of the third floor. The mini-game involves moons shooting lasers at you, and you shooting at them. To complete it perfectly, you must never be hit, and you must never hit the moons for 0 damage. In order to accomplish this, you shoot a moon if and only if it is charging a shot on you.
This is not very difficult if you keep a cool head. There's generally plenty of time for you to fire at anything before it charges its lasers and hits you, so the biggest threat is that you will prematurely fire and hit a moon for 0 damage, ruining the achievement. Make sure to wait until you see the charging animation, and don't risk firing more than once, as the moons turn around immediately and raise their shields.
If you screw up, it's much faster to die and reload the checkpoint than to reset the game, doing the whole floor over again.
Now, for the details:
There are two parts to the mini-game. In the first section there are three moons. This is best dealt with by memorizing the attack pattern. Numbering the moons left to right 1, 2, 3, they will attack like this:
3, 3, 2, 1, 1, 2
(I believe #1 and #2 leave you alone until #3 dies, but I never tested it for more than 3-4 shots. It will die in 2 charged shots). As soon as #3 dies, charge up on #2, and then charge on #1 twice, and then #2. The whole thing should take just a few seconds.
Part 2 is much more difficult, but still manageable if you have good reflexes. There are four moons now. My strategy was to memorize the opening attacks so that I could get a few charged shots in, and then bide my time until one was destroyed. This will take too long to make memorizing the whole pattern worth the trouble, and if you mess up anything the pattern will change, of course. The initial attack pattern is like this:
2, 1, 4, 3, 1
Immediately charge a shot on #2 and then partially charge at #1. Do not charge on #3 and #4, as they attack in quick succession. Hopefully #1 will die or be very close to death after its second attack. If not, keep chipping away at them all until it does. After one moon dies, the other moons will "Power Up" and turn red. For the rest of the game, they'll rapidly turn around back and forth, usually not charging their lasers. But when they do intend to charge, they will do so more quickly than before, so watch out. Just don't fall for their bluffs, and make sure that as soon as they start charging a shot you blast them. I found myself frantically moving the cursor to whichever one turned around, but again, don't fire until you're sure the moon is going to stay facing forward. The first one to fall will be #2 if you got a good charged shot on it at the very beginning.
If you made it this far, the rest should be no problem at all. You can probably start partially charging shots again since now you only need to worry about one other laser besides the one you're aiming at. Pretty soon another moon will fall and the last one should be helpless against you by itself. :)