This achievement is very easy to do and can be done across multiple worlds. In order to Discover a Biome you just have to travel to the biome so you can easily start a new world and just run around until this unlocks. There are approximately 40 Biomes currently with more coming in the Update Aquatic in Spring 2018. There are often multiple Biomes in what looks like the same area so try and be thorough with your travels. Riding a Horse around or Boating through Oceans and Jungles can really speed up your exploration.
Typically with the amount of things to do in Minecraft this will come naturally as you explore and play but for people looking to get everything quickly this one is easy to do. Happy Exploring!