Minecraft: Xbox 360 Edition

Very easy to get this super fast if you don't want to read the essay below here is a video
Load up the tutorial world. You need to make a Splash Potion Of Weakness and a Golden Apple (Golden Apple has to be made with gold bars NOT gold blocks!).

Splash Potion of Weakness - To make this you need a Vial of water (Potion brewing tutorial area),a Fermented Spider Eye (Sugar, Spider eye & Brown Mushroom) and Gun Powder (fireworks tutorial area). Add these together to make the potion.

Golden Apple - To make this you need Gold Bars (In the floor of the church at Co ordinates X: -115, Y: 69, Z: -80) and an Apple (In a chest at co ordinates X: 0, Y: 32, Z: -5)

Once you have both of these head to a village then save once night begins. exit and reload but this time on hard and I advise you turn off auto save. Let the zombies turn one of the villagers then trap the infected villager so they can't get away or hurt you.

Throw the potion at them using Then Cure them with the Golden Apple by pressing

For this achievement you'll need to cure a cursed zombie villager.

You need these raw materials; Water Bottle, Spider Eye, Sugar, Brown Mushroom, Redstone, Gunpowder, Apple, 8 Gold Bars "edit" to bars from Ditto51
Tools: Brewing Stand

First we're going to make a potion of weakness.
-Make glass vial, fill with water
-Make a fermented eyeball
-Brew them for potion of weakness

Now we're going to further build on potion of weakness
-Brew potion with Redstone
-Brew potion with Gunpowder

This will give us a splash potion of weakness with bonus

Now we'll make the gold apple with an apple and gold nuggets (made from bar of gold or harvested from zombie pigmen)

For zombie villagers you'll need to play on the harder difficulties so be careful when doing so. Next it'll be best to head to a village and wait until night in order to trap a zombie villager in a confined space or have him catch onto and follow.

Important:
Once you have the zombie villager, throw the splash potion on them and then feed them the gold apple by spamming . See video if any clarification is needed.

The other guides are helpful, especially for the part about making the potion, but there are a few things that no-one has mentioned.

First is that if you find a zombie spawner, the spawn rate for zombie villagers is 5%, so it may take you a lot of waiting to get one to spawn. That's why I advise getting a zombie to infect a regular villager to produce a zombie villager. The best way to do this is probably to destroy all the torches in a village and wait for nighttime, then lure a zombie back to the village.
The process of infecting a villager can be a little tricky. You need to trap the villager next to the zombie so that he is attacked. When he runs out of health there is a chance that he will turn into a zombie villager. Note that this chance is 0% on Easy, 50% on Normal and 100% on Hard. So you must set your difficulty to normal or higher.
Once you have infected the villager, throw your splash potion at him. As the other guides say, you make the potion by adding the ingredients to water at a potion stand in this order: Fermented Spider Eye, Redstone, Gunpowder. All these ingredients can be found in the tutorial, near the potion brewing area (of the tutorial).
You'll also need a golden apple (note that this is a regular golden apple and not a notch apple). The easiest place to find one is in the chest with the wool and dyes near the bed (in the Elytra tutorial world).
Once the splash potion hits the villager, select your golden apple, go up to him and press . This will start the curing process. What no-one else has said is that this process takes between 2 and 5 minutes, it is not instant! So ideally you should trap him somewhere while you do it. Once the time runs out he'll turn human again and the achievement will unlock.

So, in summary:
1: Preliminary work:
(i) be in the Elytra tutorial world
(ii) set difficulty to normal or hard
(iii) make splash potion
(iv) get golden apple
2: Getting the achievement
(i) have a zombie infect a villager
(ii) use the splash potion on zombie villager
(iii) walk up to zombie villager and use the golden apple to cure him
(iv) wait two to five minutes

I realise that both the Xbox 360, and this version of Minecraft aren't exactly the most played versions anymore this many years down the line, however for anyone else who, like me, is trying to clean up their older games, here's a straightforward solution.

Create a new world. As you create the world click More Options, and in the Seed for the World Generator option enter 14 to use 14 as the world seed.

Once the world loads go to the coordinates 53, 66, 33 (there should be a map in your inventory when you spawn in that you can use to see your coordinates). This will take you to an igloo at the bottom of a short drop alongside a snowy village. To make things slightly easier, on the way there collect some wood (you only need 2 wood blocks, enough to craft a wooden pickaxe).

Once at the igloo, go inside and optionally craft a wooden pickaxe using the crafting table in there. Then break the carpet tiles on the floor and you should find a trapdoor under one of the middles ones. Open the trapdoor and go down the long ladder.

At the bottom of the ladder you'll find yourself in a room with a Villager and a Zombie Villager trapped behind iron bars in front of you, a chest to your left with various food and crops in, including a golden apple, and a bench to your right with a brewing stand on it.

Open the chest and take the Golden Apple. Open the brewing stand and take the Splash Potion of Weakness that has already been created in it.

Break the top iron bar in front of the zombie villager (this is where the wooden pickaxe comes in handy). Throw the Splash Potion of Weakness at the Zombie Villager, then feed the Golden Apple to the Zombie Villager. You'll hear a loud noise, the zombie villager will start shivering, and red particle effects will start coming off him. Wait around 2 to 5 minutes, then there'll be another loud noise and he'll suddenly transform into a standard villager and the achievement will pop!

Update 20-12-2015: As of update 18.1 the tutorial world has changed and so this exact solution is no longer valid.

Since TU19 the tutorial world has completely changed! I have made a video of our process using the new world to obtain the achievement on Xbox One to help you all find the required crafting items.

To sum it up, most of the other solutions are still valid in terms of what items you need.

In the starting area, grab some food from the animal pens area to get your food and health bar up to full, grab a sword and axe from the enchantment area, grab some iron from the chest near the beacon, axe up a block of wood and make yourself a iron pickaxe (important), grab the gunpowder from the firework tutorial and fill up 3 bottles while you are there from the potion tutorial.

Go down to the water area opposite the animal pen, get the boats from the chest and head under the bridge towards the sand village, cross through to the water on the other side, you will see a ship in the distance to the right, head towards it in a boat and into the building behind it. You should be close to the bottom left corner of the map.

In here have a good look around but ultimately you need to find the library, flip the leaver next to the fireplace to open a passageway, in the room at the end is a chunk of gold blocks (hence the iron pickaxe)

All other required items can be found in this building so get brewing and crafting that golden apple! Brew fermented spider's eye, redstone and then gunpowder to make Splash Potion of Weakness+. One word of note, in this TU19 update gold bars are now required to make a golden apple so keep watch on that for future updates in case it goes back to nuggets.

Head back to the sand village you passed through and wait for the zombies to convert a villager, it is recommended to save here and reload with autosave disabled in case it all goes wrong. Not sure if it makes a difference but we reloaded the world on normal difficulty just to try and increase the chances of a zombie villager.

Once found (they look like a zombie version of a villager) throw potion and feed(?) with golden apple (both actions with LT), the achievement will unlock once you have used the golden apple, the cure doesn't have to be completed.

In the last update you can find all the ingredients for the potion in the tutorial world. For the golden apple, take gold at the beacon area then go to the village at -138, 82, 75.
Achievement will pop when the zombie become human, it can take 2-5 minutes

With the new Elytra tutorial world, this has gotten even easier.

EDIT: ...and it's apparently gotten even easier since there's a golden apple in the "main chest" near the bed room. (I think this is a relatively new addition. I think.)

Go get the golden apple. (Note that this is a SPECIAL Notch Apple, not a regular golden apple... if you need to make another for some reason, it's the one that requires 8 gold BLOCKS, not 8 gold bars.) That was most of my last solution.

Now you need a zombie villager. These spawn quite less than regular zombies, but there's a zombie spawn close to where your bed is near the Elytra. You'll follow the mine cart tracks a little, and then down the hill to the right. It's in the back of a cave. Here are the coordinates:

X: -83
Y: 70
Z: -109

In this area (both in and out of the cave), a lot of different creatures will spawn. This is where I found my zombie villager; he's a zombie but looks more disheveled. Then the fun begins.

Throw the splash potion of weakness at him, and you should see magic spirals off him. Then go up to him and press LT with the golden apple. The achievement popped instantaneously for me.

Best of luck... at least this time around, the ingredients are easier to get!