As of 1.4.1 there are 6 SLIME related achievements. 3 of them are unfortunately RNG based.
If after this guide you're still struggling with any refer to http://terraria.gamepedia.com.

- Sticky Situation:
"Survive the slime rain, where gelatinous organisms fall from the sky in droves."


Unlocking this achievement requires the 'completion' (Killing 150 slime and then also killing King Slime[terraria.gamepedia.com]) of the Slime rain event[terraria.gamepedia.com], which is unfortunately RNG based.

In order to spawn a Slime rain event[terraria.gamepedia.com] the following requirements must be met:

• One player has 140 maximum health and 8 defence (except in expert where it can occour before but with a decreased chance)
  
• The chance of it spawning in a Pre-Hardmode world is approx 4.9% (3.3% in Hardmode)
  
• If King Slime[terraria.gamepedia.com] is NOT killed the event chance is nearly doubled.

Method:
Being patient in a Pre-Hardmode expert mode world. The event is frustrating for people who're unlucky and seemingly never get the low percent chances but it will inevitably come with time.

- Slippery Shinobi:
“Defeat King Slime, the lord of all things slimy.”



To unlock the player must kill/defeat King Slime[terraria.gamepedia.com].

To summon the King Slime[terraria.gamepedia.com] the following requirements must be met:

Manual summon:

King Slime[terraria.gamepedia.com] can be summoned manually by a Slime crown[terraria.gamepedia.com].
A Slime crown can be crafted at a Crimson/corruption Altar[terraria.gamepedia.com] using a gold/platinum crown and 20 gel. To craft a gold/platinum crown you need 5 gold/platinum and a ruby.
Random/Natural summon:

King Slime[terraria.gamepedia.com] can very rarely spawn naturally if:

• It is daytime
  
• The player is located in the one of the outer lateral sixths of the map (the far left or far right)
  
• The Depth at its potential spawn point is above zero
  
• There is standard regular green grass at its potential spawnpoint (See Spawning[terraria.gamepedia.com])

OR

King Slime[terraria.gamepedia.com] will also spawn during a Slime rain event[terraria.gamepedia.com] after 150 slimes have been defeated. (Look to index for Sticky Situation for more)

- Pretty in Pink:
“Kill pinky.”


Unfortunately to unlock this tincy-wincy little achievement you have to kill the tiniest slime there ever was... Pinky[terraria.gamepedia.com].

Spawning:

• Pinky[terraria.gamepedia.com] has a 1/180 (0.56%) chance of spawning in place of any standard slime
  
• The more slimes there are the more likelihood of a Pinky. Slime rain[terraria.gamepedia.com] is the best time to find it.

- Dungeon Heist:
"Steal a key from dungeon's undead denizens, and unlock one of their precious golden chests."

Unlike previous achievements this achievement isn't strictly 'slime' related. To unlock it you must open a Dungeon chest[terraria.gamepedia.com] using a Golden Key[terraria.gamepedia.com].

Obtaining key

• Killing a Dungeon Slime[terraria.gamepedia.com] (100% chance)
  
• Look to Golden Key[terraria.gamepedia.com] for other non-related slime methods...

- Just Desserts:
"Defeat Queen Slime, giving the coup-de-grace to the sovereign of all that jiggles."


To unlock this achievement Queen slime[terraria.gamepedia.com] must be defeated.

Summoning
To summon Queen slime[terraria.gamepedia.com] the player must consume a Gelatin Crystal[terraria.gamepedia.com] whilst in the hallow.

To obtain a Gelatin Crystal[terraria.gamepedia.com] the player must go to the underground hallow and look at the undersides/sides of pearlstone blocks for places where crystal shards typically spawn.

Queen slime strategies:

Ngl im too lazy to go too in-depth with intricate fight preparations but i will link the WiKi: Strategies[terraria.gamepedia.com]

- Gelatin World Tour:
“Defeat every type of slime there is!”


This achievement is across your entire terraria account. You only need to have killed one once on any of your terraria characters for it to be counted. Tips: use your Bestiary[terraria.gamepedia.com], and write down all defeated slime.

There are 24 different slime required for this achievement, but there are a lot of slimes that don't count... here is a nice checklist of those that do:

Blue Slime[terraria.gamepedia.com]
Spawns in forests during day or from slime statue.

Green Slime[terraria.gamepedia.com]
Spawns in forests during day or the Underground[terraria.gamepedia.com].

Red Slime[terraria.gamepedia.com]
Spawns in the Underground[terraria.gamepedia.com] layer.

Yellow Slime[terraria.gamepedia.com]
Spawns in the Underground[terraria.gamepedia.com] layer.

Purple Slime[terraria.gamepedia.com]
Spawns in the outermost third of the map and during the Slime Event[terraria.gamepedia.com].

Umbrella Slime[terraria.gamepedia.com]
Spawns in forests during rain.


Sand Slime[terraria.gamepedia.com]
Spawns in the Desert[terraria.gamepedia.com].

Jungle Slime[terraria.gamepedia.com]
Spawns in the Jungle[terraria.gamepedia.com].

Spiked Jungle Slime[terraria.gamepedia.com]
Spawns in the underground Jungle[terraria.gamepedia.com].

Ice Slime[terraria.gamepedia.com]
Spawns in the Tundra[terraria.gamepedia.com] (Snow biome).

Spiked Ice Slime[terraria.gamepedia.com]
Spawns in the underground Tundra[terraria.gamepedia.com] (Snow biome).

Lava Slime[terraria.gamepedia.com]
Spawns in the Underworld[terraria.gamepedia.com].

Mother Slime[terraria.gamepedia.com]
Spawns in the Cavern[terraria.gamepedia.com] layer.


Baby Slime[terraria.gamepedia.com]
Spawns after killing Mother Slime[terraria.gamepedia.com], which spawns in the Cavern[terraria.gamepedia.com] layer.

Black Slime[terraria.gamepedia.com]
Spawns in the Cavern[terraria.gamepedia.com] layer.

Dungeon Slime[terraria.gamepedia.com]
Spawns in the Dungeon[terraria.gamepedia.com].

Pinky[terraria.gamepedia.com]
1/180 chance for it to spain in the place of any other slime. More likely during Slime Rain[terraria.gamepedia.com].

King Slime[terraria.gamepedia.com]
Spawns in the outer thirds of the map, after killing 150 slimes during a Slime Rain[terraria.gamepedia.com] and after consuming a Slime Crown[terraria.gamepedia.com].

Hard Mode Slime

Toxic Sludge [terraria.gamepedia.com]
Spawns in the Underground[terraria.gamepedia.com] and Cavern[terraria.gamepedia.com].

Crimslime[terraria.gamepedia.com]
Spawns in the Crimson[terraria.gamepedia.com].

Corrupt Slime[terraria.gamepedia.com]
Spawns in the Corruption[terraria.gamepedia.com].

Slimeling[terraria.gamepedia.com]
Spawns after killing Corrupt Slime.

Slimer[terraria.gamepedia.com]
Spawns in the Corruption[terraria.gamepedia.com].

Illuminant Slime[terraria.gamepedia.com]
Spawns in the Underground Hallow[terraria.gamepedia.com].

Rainbow Slime[terraria.gamepedia.com]
Spawns during rain, whilst the player is in the Hallow[terraria.gamepedia.com], on non-hallowed blocks.