This is probably the second most irritating collectables achievement in GTA V, second only to the Stunt Jumps.
There are a total of 30 barrels of nuclear waste to collect. First, you must buy the old dock. This can be found about three-quarters up the west of the map, and will be marked with a icon indicating it is for sale. It can be bought by any of the protagonists. You must have completed the Merryweather Heist main story mission before being able to buy it, and it costs $250,000. You will make this money back simply by collecting the barrels (you get $23,000 for each one you collect).
A submarine spawns at the end of the dock. Get in it, and you will get a brief tutorial on how to use it. Very usefully, you will have an app on your in-game phone that broadcasts each of the barrel's locations when you get near them. Thus; you don't need to be able to see the barrels to know where to find them.
I would suggest starting by heading north from the docks, going along the top of the map, then down the east side, and finish just to the south-east of Los Santos, then leave the sub, and get a taxi back to the docks, before heading down the west side. This seemed to me to be the most efficient way of doing it. They tend to appear in clusters, so chances are, if you've found one, you'll find a few more.
A couple can be found close to crush depth (492ft). If you go much deeper than this, the submarine will implode and the protagonist will die. Be careful about damaging the sub. As the first video below shows, you can break the sub if you are careless with controlling it, once again, killing the protagonist. If that does happen, I suggest getting a cargobob, and carrying the submarine that way; it saves you a lot of time.
If you're having trouble picking some of the nuclear waste up, Milki Tea
suggests trying this:
would just like to add some info on a texture problem that prevents you obtaining some nuclear waste. the Coral on seafloor spawns randomly when you load game up, sometimes it loads ontop of nuclear waste and you cant get to it. simply do a quick save and load game up again until coral is clear of the waste can if you encounter this as i did.
The videos below show you the location of each nuclear waste barrel. They were found on the Xbox 360 version of the game, but the locations are the same on the Xbox One.
Additionally, the Xbox 360 version had a feature that told you how far away depth-wise you were from the nuclear waste. This feature is missing from the Xbox One version of the game. The videos should give you a guide of how deep you should go to find the next barrel.