The giant crystal is found in the Snow biome.
Makes sure to talk to the NPC before starting to mine it, the achievement may glitch. The miner will progressively mine at the crystal until it breaks as soon as you spawn the island so talk to him ASAP.
The crystal can take quite a while to mine if you don't have very much power and upgrades. I recommend tackling this biome a bit later, at least until you have the Thunder Rod from the desert temple. This takes mining down considerably and will probably take a minute to mine. The dwarf guy in the island will also mine slowly for you if you want but doing it yourself will be faster.