Minecraft: Xbox 360 Edition

EDIT: It seems that the coordinates in this solution and comment section no longer work since a recent update and have been changed, will try to do my research and give you the new location (if it exists).

You can find a chest with a diamond in the Sand Temple (X: -229, Y:66, Z:41) in the Tutorial World. Head west and you'll spot the temple near a village by the shore, enter from the main entrance and you will see on the floor a blue wool block and some orange wool blocks, dig down one of the ORANGE wool block and you'll fall down in a pit with a dynamite trigger pad on the center, without stepping on it, search the 4 chests that are mounted on the walls and you will find a diamond there, just take it in your inventory, toss it on the floor and pick it up again for the achievement.

Now, if you want to get

dig some sand blocks and return to the surface, get back to the village that you spawn in and exit and save the game. Now connect a second controller or call a friend and throw the diamond to your second player to pick it up.

EDIT:
Thanks to DefiantZero and Zordnil for pointing this out:

If you already have a world where you have diamonds, you can get this achievement if somebody throws one at you for the other achievement.

There is also easy access to diamond in the tutorial world at co-ordinates X:-158 / Z: 329
(Again, thanks Zordnil)

Confirmed on 29/05/15

Old Solution:

Firstly you will need only an Iron Pickaxe, which is created using 3 Iron Ingots and 2 sticks.

Then you will need to mine down to level Y16 or lower, this is because Diamonds only spawn at this level,
(HINT: You can tell what "Y" level you are on by looking at your map, it displays on the top)

Personally I have the most success mining at Y12, though your mileage may vary
The most common way to hunt for Diamonds is to "Strip Mine" this is where you mine into the rock in 3 block intervals as below

XCCXCCXCCX
XCCXCCXCCX

X: Mine Here
C: This is stone

It shouldn't take you too long to find diamonds using this pattern.

This video is not mine, but it should help you if my explanation isn't so useful

If you already have a world with a diamond in it simply drop it and pick it up.

Simply start the tutorial, go down to where the beacon is, take the iron out of the chest, break any wood, make wood planks and then sticks, go to the crafting table and make an iron pickaxe, go back to the beacon area, break a diamond block, go back to the crafting table, turn the diamond block into diamonds, throw one on the ground, and pick it back up.

It the tutorial map if you dig to coordinators X-191, Y 13, Z 240 there are two blocks of diamond ore just sitting at the bottom of a abandoned mine shaft surrounded by lava columns and water columns

Since the tutorial world always changes, here is my suggestion. As soon as you got an Iron Pickaxe, start digging down until you reach Y: 11 (Y coordinate defines the up/down axis. You can use your map to see your actual coordinates) Start digging in a straight 1x2 corridor until you find some diamond orbs. It might sound crazy, but believe me, most Minecrafters using this method to find diamond fast. If you accidently happen to find a cave, start going down and look for lava ponds. Diamond orbs are usually generated near big lava ponds.