Minecraft: Xbox 360 Edition

THIS SOLUTION NO LONGER WORKS SINCE IT WAS MADE FOR AN OLDER TUTORIAL WORLD

For this you will want a sword from wood to diamond. It is preferable to have at least an iron or diamond sword though.

You'll also need a potion of strength which is a bottle of water, nether warts, and blaze powder. If you add glowstone dust into the potion it does 260% damage vs the regular 130% damage.
Also pops if you havent already
Adding the glowstone dust allows you to use any sword of your choice from wood to diamond. However if you use a regular strength potion you will need iron or diamond swords.

Now drink the potion and kill spiders, skeletons, zombies, slimes, creepers, or any of the farm animals and it should pop

Enchanted weapons don't work for this as far as I've seen and people have noted. However recently with the new updates I have had one person kill five different things for one achievement and another use an enchanted sword and get it within three kills. Just keep smacking stuff till it pops.

Super easy to get this one. Load up the Tutorial World and make a Diamond Sword and Potion Of Strength II (Details on how below) then find a Villager, take the potion and bash the crap out of them with the sword.
Diamond Sword - Grab the 5 iron bars out of the chest near the beacon and make an Iron Pickaxe. Go to the South West corner of the map (to the building with the pirate ship outside). On the first floor, get the block of Diamond from behind the fire in the Library.

Potion Of Strength II - Grab a vial from the potion tutorial area and fill it with water. Head to the building in the South West corner of the map (the building with the pirate ship outside). Go to the second floor and head to the room filled with chests full of goodies. Go up one of the ladders and find the chests with the Blaze Powder, Nether Wart & Glowstone Dust in them. Use the brewing stand to add all ingredients to the water vial.

This solution works as of the Update Aquatic

In Minecraft, half a heart represents one hit point. So this achievement requires 9 hearts = 18 hitpoints of damage.

As far as I know, this will pop when you do that much damage to any mob, or another player, even if that's more than their maximum health. I did it on a skeleton.

Oversimplified explanation of Minecraft's damage system

Your base damage dealt is based on the attack points of whatever weapon you're using. You can add points by using a better weapon, enchanting your weapon, or using a potion of strength. Dealing a critical hit, which for melee weapons means attacking while falling, increases your damage by 50%. Attacking several times in quick succession will decrease your damage.

Your best bet for this achievement is to boost your attack into the teens, then take a single swing while falling to boost you up to the required damage. The part of a jump where you come back down counts as falling.

How to prep in the Tutorial world

In the agricultural area, there's a villager house with an anvil. In the chest outside, you'll find several Bottles O' Experience. Take these.

Go out into the main water area, and follow the potion signs on the left. Keep going up and through this area; you need Glowstone Dust, which I believe you can find in a chest in the fireworks area. Take that back to the potions room. Take bottles, Nether Wart, and Blaze Powder from the chest. Fill up three bottles and place them in a brewing stand, then use one Blaze Power for fuel, then one each of Nether Wart, Blaze Powder, and Glowstone Dust in that order to brew three level II strength potion, which when drank will increase your attack by 6 points for 1:30 minutes. Hold onto these for later.

Directly above the potions area is an enchanting area. Before you go inside, go to the Elytra area and grab several more Bottles O' Experience from the chest. Then go to the enchanting area, grabbing experience bottles from that chest as well as Lapis Lazuli and a diamond sword. The Bottles O' Experience are thrown potions, so smash them all against the wall and pick up the resulting experience orbs.

Disable autosaving and save your world. Go to the enchanting table and place the diamond sword in. You probably won't be a high enough experience level to get whatever enchantment is in the third slot, so look in the second slot. If it's something like Smite or Sharpness at at least level II, take that; if not, quit your world and reload until you get it.

Your sword should now be at roughly 8 attack. Find a mob, take your potion to give you another 6 attack, then jump and take a single swing at the mob on your way down to unlock the achievement.

Copied from my Xbox One solution

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