This guide only covers the checkpoint-achievements as the level-achievement is next to impossible to achieve.
At the end You can find a more in-depth guide stating my reasoning behind my choices.

Select Yuki, The Beach - Normal.
Jump to the right until reaching the grey heart - move towards it, making it red and then stop jumping so You die.
Once respawned, Quit to Main Menu [ESC] and repeat!

Resolution
Regardless of your chosen resolution - Your character will be too big for the playing area, there is no way of getting around that.
Inputs
I have only two actions operating properly - jump left and jump right. You will only need to jump right.

Gamemode
Select Story-mode.
Character
Select Yuki, due to being visually smaller. Unsure if their hitboxes differ.

The Heaven & Battlefield
Completely broken, regardless of input You will suffer imminent death by the trollface in a hot-air-balloon. SKIP
The Beach
Now we're talking - finally a playable level! This level has two components - starfishes & falling coconuts... bowling-balls?
The starfishes are extremely hard to dodge - if You're in their path, expect to die/restart. They bounce off the walls!
The coconuts bowling-balls will fall in even intervals, if needed, You can jump to the left to keep Your distance before crossing its path.
The Universe
Somewhat playable - featuring: Meteors.
The meteors falls at a constant rate - however, unlike the coconuts bowling-balls - the meteors fall diagonally. These are quite difficult to dodge.

Normal.

Jump to the right at a consistent rate to stay at the middle of the screen. Once You see a grey heart - move towards it making it red.
Once it is red - stop jumping, fall of the screen - killing You.
Exit to main menu [ESC] and repeat for all checkpoint-achievements.

So... after all this time, I finally managed to reach the end of a level. Something I tried to establish even at the creation of this guide was whether or not the resolution changed the difficulty of the game. My findings are that there is a significant change in the behaviour of the moving enemies (starfishes, knights and the troll-balloon). They appear to collide and get stuck somewhere along the level, making it far easier to simply dodge static objects. I also believe moving objects like meteors, coconuts and balloons are falling at a set pixel-speed regardless of resolution, so it is advantageous to play on a smaller resolution, as your character's hitbox is smaller, making it easier to pass these obstacles.

Thus, I would recommend the resolutions 1360 x 768 (I managed to get it first try with this one) or alternatively 1280 x 720 (Almost got it with this one). It's a question of preference, as the aspect ratio dictates how well you can see the objects ahead of time.

I took the advice of "JustWater"'s comment to play the level Heaven with Yuki. However, playing on certain resolutions other than the ones I mentioned above can result in the balloon man appearing, rendering it impossible to even begin the level

Don't ever under-utilize jumping backwards, it is essential to circumvent unpassable scenarios

It is highly based on luck, as the main obstacle is to dodge the combination of the knight and the starfish - both of which appear to start at random locations, which could lead to you encountering them both at once. Problem of this is of course, starfishes occupy the upper half of the screen, while the knight can move up to 3/4ths of the lower screen.

Also, don't worry. The achievement does not pop immediately. I began another run of the same level before closing it. I have no idea if that made any difference, but I got the achievement when I exited the game.