Minecraft: Edition

Start a new tutorial world, Press B to skip the tutorial on how to play and then run over to these coordinates;
X: -302
Y: 70
Z: 106
There will be a chest with a Iron sword with Smite V enchantment which increases damage to "undead" mobs (skeletons, zombies and zombie pigmen).
Now simply wait until night and kill a zombie or skeleton with the Smite V sword!
Extra: If your lazy like me and cant wait until night to get the achievement, there is a dungeon with a zombie spawner at these coordinates;
X: -41
Y: 60
Z: 8

Title Update 19 Tutorial World Compatible
Updated damage information - 11/06/2015

Start up a new tutorial word and skip the tutorial. Firstly, go over to the workshop located next to the beacon. Open up the chest next to the anvil and get the stack of experience bottles. Also get the 5 iron from the chest next to the beacon.

Next, go to the enchanting room and get the other stack of experience bottles from the chest there, throw both stacks on the ground to get level 30. Next, make an iron sword from the iron you collected earlier.

Disable autosave and enchant your iron sword, keep quitting the game until you get a sharpness or smite enchantment of either 4 or 5. Wait until night time and kill a mob, read the information below to see which one you can kill. You may need to critical hit a mob, which is hitting it while falling.

We'll focus on the four main aggressive mobs in the overworld. Each heart is two points of health, so a Skeleton has 10 hearts. Sharpness does 1.25 damage per hit on any mob. Smite does 2.5 damage per hit on undead mobs. Bane of Arthropods does 2.5 damage per hit on spiders.

Mob Health:
Skeleton - 20
Spider - 16
Zombie - 20 (2 Armor Points)
Creeper - 20

Damages:
Iron Sword - 7
Iron Sword w/ Critical Strike - 10.5

Enchantment Damages:
Sharpness 4 - 10
Sharpness 5 - 12.5
Smite 4/Bane 4 - 20
Smite 5/Bane 4 - 25

Using a advanced strength potion which gives 260% Melee damage and a Diamond sword with a minor damage upgrade from the enchantment tables/books (I had sharpness II) on a monster will net you this achievement.

To make the Strength Potion II you will need.

2 x Blaze Rod
3 x Sand/Glass
3 x Cobblestone
1 x Nether Wart
1 x Glowstone Dust
1 x Crafting Table
1 x Furnace + Fuel
1 x Water Source
10 x Obsidian + Flint and Steel (if nether is required for materials)

Create a Brewing Stand from the 4th box in the Structure menu by combining 3 cobblestone and a blaze rod at a crafting bench. Blaze rods can be acquired in Fortresses from blazes in the Nether (get 2 rods to save time).
Use the 2nd blaze rod to create blaze powder from the Food tab either by pressing X or using a crafting table.
Use 3 sand in the furnace to create glass.
Use 3 glass to create glass bottles from the tools and weapon tab in the crafting table.
Find a water source block and fill the glass bottles with water.

Now return to your brewing stand. In the brewing stand menu.
Add the Water bottle to the brewing stand.
Add Nether Wart to a Water Bottle to create an awkward potion.
Add Blaze Powder to the awkward potion to create a strength potion.
Add Glowstone Dust to the Strength Potion to get a Strength Potion II

Drink the potion, once active red bubbles should emit from the player. Simply hit an enemy (I used a monster not sure if it works for animals) with the low enchanted sword for the achievement.

Further potions can be found here: http://minecraft.gamepedia.com/Brewing

If any steps are unclear, let me know I'll try and assist.

Diamond Sword - Bane of Arthopods IV & Fire Aspect II

I used this set-up to get the 'Overkill' achievement. All you need to do is find a spider and hit it. It'll one hit kill the spider and the achievement will pop. A normal diamond sword will do 4 hearts of damage. The Bane of Arthopods coupled with the fire aspect allowed me to do the other 5 hearts of damage.

The best way to get this type of weapon is set up an enchanting table that is coupled with libraries. Get your level up to 30 and then save your game. The next step is to try and get a diamond sword with these abilities on the weapon (Sharpness can be beneficial but the main attribute needed is the Bane of Arthopods). Once you get a weapon with these abilites, go and find the nearest spider and you should be able to pop this achievement!

Credit: Whaticles for creating the weapon itself and hosting the game.

Here is a great video for using the Tutorial in 2017, super easy! Video is for earning trophy on PS4, so when he says hit "circle" it translates to the "B" button for us. I used the ideas from this to quickly make 5 or 6 swords until I got a couple high level ones perfect for a few different mobs, combined with the Strength 2 potions that are easy to make w/ all the ingredients waiting for you!

Let me know if I can make this solution better. I know there are a ton of great solutions above, I just found something more relevant to me with the latest version of the tutorial world and thought I'd help others.

This achievement can be unlocked easily by enchanting an iron/diamond sword with the Smite V enchantment.
Once you have enchanted your sword, simply defeat any zombie or skeleton and this achievement should pop.
The book of enchantment can be obtained easily in the tutorial world along with an iron sword if you want to get it quickly.

Here's a video to help:

For this achievement, I used a Diamond Sword, which was enchanted with Sharpness IV, Fire Aspect II and Durability III. Only one of these achievements matters, since it's all about Damage.

The sword deals around 7, 7-5 hearts of damage, so there was no way I could get the cheevo to pop without a little thing almost everyone forgets: Strength Potions.

Strength potions can, when brewed correctly, deal twice more damage as the normal potion. This means that with this potion on you, you have 8 minutes to find a passive or agressive mob with 10 or more hearts. If you hit the mob with a critical hit, the cheevo should pop!