This is by far the easiest achievement of all. Basically all you have to do is kill one guard. The easiest way to do this is by sneaking up on the guard from behind and performing a stealth kill by completing the indicated quick time event.
In order to do this, you need to own a a sword, which you will pick up around the start at the first level.

This achievement is rewarded when saving Azai, which is also the only way to complete the first level 'Ink & Dreams'.
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This achievement is rewarded when completing a total of 25 stealth kills. There are several ways to perform a stealth kill. The most likely one you'll use is the 'Silent Assassin' attack, which is performed by sneaking up to the victim from behind. Other stealth kills can be unlocked with 'honor', which is rewarded when you either score enough points, find scrolls or complete various objectives in a level. Buying stealth kills allows you to kill a target from new positions, such as from a hiding spot, from below and from above.
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To beat the trickster achievement you have to distract 25 guards using environmental objectives. The most common way to do this is by breaking lights using your bamboo darts. The sound that it emits will distract the guards. You can see if the sound waves will actually reach them by selecting a light in aiming mode and by looking if the blue circle surrounding it reaches the guard.
Another way to distract guards is by hitting gongs instead of light, which requires the same tactic as with lights.
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To gain this achievement you need to lure another guard into a kill by using a dead body. A good way to do this is by killing a guard, leaving it be and wait for another guard to come near. If you are able to hide somewhere above the dead body, you can use the Bat's Prey stealth kill to jump upon a guard if he comes near the dead body. Make sure the dead body is in a position where another guards flashlight will cross over it. This is easiest to do when the guard you plan on luring walks a path, so you can place the dead body there when he walks the other way and get into position before he turns around.
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The Mercenary achievement is rewarded for killing corporal Kelly at the end of the fifth level.You can kill him by dropping a big electric platform on him, of which you need to cut the wires that hold it using bamboo darts. First though, you have to disable the lasers be sneaking behind the snipers. It's recommended that you stealth kill them too, so you will get additional points and they will be no threat anymore.
Once you dropped the platform on Kelly he's disabled. From then on all you have to do is kill him, which should be no problem as he can't fight back at that point.

In order to get this achievement you will need to terrorize a guard. You can do this most easily by throwing a guard you have already killed at your next victim. He will then start shooting aimlessly. The achievement is earned if he shoots one of his allies. You will also get 700 points for 'Friendly Fire'.

You get this achievement if you manage to beat a level, without getting detected once. All you have to do is avoid getting seen by any guards, which is a piece of cake if you try to avoid them alltogether. If you don't let them hear you (by running near them), don't get in their vision and avoid light, you should have little trouble getting this achievement. Especially on the easier levels.

In order to get Crimson Haiku, it's best that you unlock most, if not all, available stealth kills.Possible stealth kill combinations available in most levels are:

Silent Assassin; the regular stealth kill.
Bat's Prey; jumping upon an enemy and killing them before you land.
Emperor's Abyss; dragging a guard down from a vent.
Twilight Gate; kill a guard standing near a door, from the other side.
Hangman's Hymn; use your grappling hook to tumble down and kill the guard when near enough.

At level eight, you can get this achievement by stunning an Elite guard. The method adviced by the game is pulling the lever activating the electric wires when he walks by. If you do this correctly, it's an easy kill.

I will hopefully add more in the coming weeks. I hope I have informed you to this point, leave any questions below.