You'll first need to play a few main story based mission to unlock the Deep Dive Syringe. Once this has been unlocked and you've made one with the required coloured plants (2 Blue and 1 Green), head towards the sea. You're now looking for an area with the darkest blue to it. Head to these coordinates for an area opened up early to you - X: 709.1, Y: 835.6.
Now equip and use the Deep Dive Syringe and dive down to the bottom. Your best bet here is to press to swim faster so that you don't drown. Once you've reached and touched the bottom of the sea, the achievement will unlock.
Search for a dark blue part of the ocean on your map. These are the deepest parts of the ocean. You need to activate radio towers to uncover the map. After reaching such a point in the ocean you need to dive down until you hit the ground. Click the left stick to swim faster.
You can get this in a bay on the south east of the first island. There is a shark relic that appears as though it is just off the beach in the shallow water. In reality, it is at the bottom of a deep shaft (their is also a chest at the bottom). Swim straight down and get yourself a cheevo and some loot :)