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