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:
4) 153 898 937 729 1000 229
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.