How is it possible to get killed by the Green Slime?

You're going to need to be quite creative with this one, but it can be done. The good news is, if you manage to get a green slmie to hurt you, it seems to do a lot of damage. At least 5 hearts worth.

Here's the setup and explanation:

First of, you need a way to make the green slime move. Almost no enemy can hurt you, unless they move into you. Scroll of Fear will make the slime move without hurthing it.
Secondly, you need to move to have the slime move towards you. There are a couple of ways to do this.
Scenario A: Confuse
When the slime gets confused, it will move in the opposite directions. This means, in stead of moving away from you (under the Fear effect), it moves towards you. There are a couple of ways to cause confuse

The most reliable way to get a slime confused, is to play Dove (unlocked by beating Zone 2.) Her weapon is the Flower that causes Confuse on all the enemies hit. Of course, playing Dove makes finding the Scroll of Fear less likely, as shops will only carry two items instead of three.
Alternatively, you need a Confuse Trap. The Feared Slime will need to move onto this trap at which point it will swiftly change direction and berge right into you. (May not be viable as Confuse Traps and Green Slimes don't seem to spawn on same floors.)

With all this done, you only need to stand still and let the green, slimy nature take its course.

Scenario B: Bounces (highly unlikely)

If you're really lucky, you'll find the bounce traps in a configuration that will bounce the frightened slime on you. These may be rather rare, tough, as you need the slime to bounce one square behind its original position.

I've tested this in the level editor mode and these both work reliably. However, it seems you can't get the achievement this way. You'll just have to play the game normaly and hope to get what you need.

Who knows, maybe someone will find a seed that will have these setups easily attainable. According to the developer, you can't unlock achievements with seeded runs.

Based on the possibilities described by DJ Pirtu, I have found another way.

The AI of the green slime is very simple : do not move.
To make him move, we have to teach him an AI, which is doable by using the scroll of fear, I don't think there is any other way to alter an ennemy's AI.
If a green slime move to you, he proceeds to hit you, like any other mob. Fortunately, he has enough strength to kill a 5-hearted player (to be tested with 10 hearts equipped with the full-armored set).

Based on that, you can use local co-op.
Put the two players in each side of a green slime.
P1 Slime P2

One of the players use the scroll of fear, and the green slime will flee him, causing him to run on the other player and instant-kill him, leaving the other player alone.

I don't know if it unlocks the achievement, (I already did thanks to DJ Pirtu) but at least, the green slime killed my character.