How to you get away with your sixth murder

• Get a gold weapon (preferably the longsword) from one of a blood shop (they're pretty common in 1-1). According to the wiki, what they do is drop 1-3 extra coins per monster, although I had the feeling that they would increase gold drops by significantly more than that. The gold longsword does costs 1.5 hearts to buy (leaving you with half), but it's a perfectly reasonable and reliable weapon to win the game with.
• The ring of war increases your damage, your knockback and the amount of enemies spawned in future floors. More enemies means more gold.
• The ring of shadows lets you get items from all shops for free. That can be very helpful early on.
• The crown of greed doubles all gold pickups, but you lose 1¢ per turn. You can get it by killing the shopkeeper, which is easier said than done. As a side effect, this kills the shopkeeper on all floors, which means shops will be empty.
• Ballet shoes mean you will only lose your coin multiplier when you take damage.
• If you can find a green scroll of need and you have precisely 0¢, you'll get 500 for free.
• The yellow scroll is easier to take advantage of: use it when your coin multiplier is 3 to get 150¢.
• If you have an obsidian shovel, get a coin multiplier of 3 and use a purple scroll in the neighbourhood of a shop. While gigantism persists, you can dig down gold walls for 10¢ apiece.
• Even just a regular pickaxe, however, leaves you with lots of leftover bombs. Use bombs on the inner corners of shops for 50¢ a pop. That makes transmogrifying your basic shovel easily worth the price.
• The shrine of war makes all enemies in future floors into their hardest variants (e.g. all skeletons will be black and all bats red). Harder enemies means more gold (but those black riding skeletons are quite annoying to deal with).
• Arena runes (the black ones) give you two more minibosses to defeat, with the associated gold.
• Wait for zone 4 to be released. So far it seems like you really have to get the crown of greed to get above 6,000¢ by the end of zone 3. When Zone 4 will be released, you can probably reach that milestone by simply using a gold weapon, killing all enemies and a ring of war.

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.