Minecraft: Xbox 360 Edition

A simple way to get the achievement is creating multiple worlds from different seeds. Below there's a video showing all biomes you need for the achievement.
List of biomes:
How to insert the seed:
When you create a new world you can go on "More Options" and insert a seed or leave it blank for a random seed. You should only put once the seed and hit back to create the world, if you select "more options" again the seed will be blank.

The seed needed:
1) 2
2) 14
3) 30
4) 153 898 937 729 1000 229
5) 32478
6) 902 963 153 450 15 956

You don't need to go too far with those seed, most of them you re spawn in biomes close to another, run around a little should be enough. For more detail, see the video below.
If you still don't get the achievement you can try the tutorial and this site to find various seeds of biomes: http://chunkbase.com/apps/biome-finder

All credit to the video owner.

Simple! If you don't want to read the essay below here is a video.
There are a total of 23 Biomes:

• River
• Frozen River
• Taiga
• Taiga Mountain
• Cold Taiga
• Cold Taiga Mountain
• Extreme Hills
• Plains
• Ice Plains
• Forest
• Swampland
• Swampland Mountain
• Mushroom Island
• Mushroom Island Shore
• Beach
• Cold Beach
• Jungle
• Jungle Mountain
• Jungle Edge
• Desert
• Ocean
• Nether
• The End

Method 1 - Most map will contain almost all of the Biomes in the game. Simply set the game to peaceful and walk around unlocking the map.

Method 2 - You can make a new world, then make a copy of the save then load up the copy in creative mode. This will not disable achievements in the original save and means you could fly around looking for the Biomes listed and write down their co-ordinates from the map. Then simply load up the original save on peaceful and walk to the co-ordinates.