Beat the second boss without dying.
Note that this achievement must be unlocked on the regular version of the first stage; it cannot be unlocked on the mirrored version. Also note that if you do die during the boss fight, in order to attempt this achievement again, you must restart the stage from the beginning.
This fight consists of two halves. The first half sees you dodging blocks/spikes that shoot across the screen, and the second half has you dodging the boss. The first half starts off with a simple matter of putting yourself in the gap between the blocks to avoid the spikes. When all five lanes are occupied by blocks, know that the top blocks shoot down, there is a pause for a second, and then the bottom ones shoot up. What this means is, for all the sections where all five lanes are occupied, you need to be above a block on the bottom but next to a lane with a block on top. Right after the top blocks shoot down, move into the now-open lane next to you to avoid the blocks shooting up from the bottom.
After that, the second half starts, which consists only of dodging the boss, who shoots up and down the screen at regular intervals. The walls will also close in, so pay attention to the grayed background to see where the walls will go, and make sure you stay in the middle (except for the one instance where the wall splits the screen in half; choose any side, but then get back to the middle after that instance). Avoiding the walls is pretty simple to do. Avoiding the boss isn't difficult once you get the rhythm down. Using the music in the background helps tremendously here. As soon as the boss enters the screen, wait for him to shoot down, and as soon as he does, start counting on the beat. I broke it down into counts of four beats, which you could either say in your head or even out loud if it helps. The pattern/timing looked like this for me, where [shoot] is the boss shooting up or down: (start counting right after the boss shoots) 1 - 2 - 3 - 4 - 1 - MOVE - [shoot] - 4. Obviously move where it says MOVE. This count of 8 beats just repeats over and over. If it helps, what might be easier is to start counting on the beat immediately, but count from 1 to 8, and then just repeat from 1 to 8 over and over. Then, the first time the boss shoots, make note of what number he shoots on, and then for each repeated cycle, move to the other side on the beat just before he shoots. So, if he shoots on your count of 5 each time, move on 4. While counting, just stand still as close to one side as possible, and then move to the other side right before he shoots.
I highly recommend watching the video in the "11-00110001" achievement solution to see what the patterns are for this boss fight. There isn't anything difficult in this fight, so as long as you're prepared for the first half, and can keep a rhythm for the second half, this achievement shouldn't take more than a couple attempts.