First you'll need to travel to the nether - I can't confirm that all the new biomes will spawn in an existing world as I did this using a newly created world - you might want to create a new world too and go the extra mile to get some diamond so you can make a diamond pick axe to get the obsidian you'll need to make a portal. There seem to be many more broken portals out in the open in the normal world now, so you shouldn't have much difficulty finding one - there will be a chest there that might have some obsidian inside and just in case you don't already know this, you can also mine the four corners of an existing portal to get four obsidian as the formation below makes a valid portal (where x is empty and o is obsidian).
There are five biomes in the Nether that you need to visit to unlock this trophy:
1. soul sand valley - brown sandy blocks with some burning blue flames, a few bone block statue like features
2. warped forest - blue trees, blue grass - you can't miss it
3. crimson forest - red trees, red grass - you can't miss it
4. basalt delta - ash coloured blocks and darker brownish blocks rising up in columns of different heights, with lava interspersed (not big lakes of lava, sometimes can be just one square here and there, like a checkerboard)
5. nether wastes - the old biome we all know and love, with lots of lava, nether fortresses, and glowstone blocks
It may take a while to find them all and the nether is sometimes a bit hard to traverse. Take lots of gravel or sand with you as it's useful for getting yourself from a high point to a low one when there doesn't seem to be any obvious path down.
Also, you can do this on peaceful - I say this because I was overrun with so many mob spawns, even on easy, that I found it very hard to deal with them while carefully making my way across the nether.
If you've searched for a while without success finding all five biomes, you could return to the normal world, take 10 obsidian blocks and a flint and steel with you, and travel 500 or so blocks on both x and z away. Build a nether portal and go back into the nether. Hopefully you'll be some distance away from your first portal and in a different biome (or at least closer to one). This is a quick way to jump around the nether. I found it useful to have coordinates displayed on screen and write down portal locations so I could find them again.
A seed you could try is -1337710146. Coordinates for the biomes are:
1. 935 60 -16
2. 323 33 -60
3. 215 60 -72
4. 593 43 -135
5. 206 50 -37
(credit to Blocky on youtube for most of this). I can't confirm this will unlock the trophy as although I found the biomes in these locations I didn't get the trophy in this world.
This trophy is glitchy for some players so you might find you go to all five biomes and don't unlock the trophy (this happened to me). In this case, follow the instructions in this video by R4ND0M G4M3R - worked for me.
Happy nether exploring!