Get your Diving gear (Diving suit, Diving Goggles, and Rebreather. This will require having a Diving license. And ofc your gps, backpack and if necessary NV-Goggles)
Then you go to a market and buy an excavator.
If you are alone:
Get a boat and sail to the shipwreck, dive down until you see the actual wreck. Then go into your Y menu and use the excatator.
If you are full you can sell it at the antique dealer #profits (Correct me if I'm wrong I never got anything besides old-gold bars from those HM wrecks)
If you are with two or more people:
Get a Hummingbird (You can jump off the sidebench and you can climb back up that way to) Make sure you have a good pilot with you.
Fly to the location of the wreck and let the pilot hover a couple of feet above the water and let the Diver(s) jump out.
Swim down and excavate. Swim back up and get back into the Hummingbird.
Then sell it at the correct market.
I prefer method #2 since that's faster.