Start off by going to the objective marker, which will bring you to the bridge. You will be waiting a little bit before you see the smuggler, who will be coming on a boat, so you’ll need to quickly get onto the hovercraft that’s on your map and give chase. If you haven’t noticed by now, these smugglers are always accompanied by another vehicle, so watch out for them.
When he lands, kill him, loot his key and then clear the area of the rest of the enemies. Go inside the vault and loot the diamonds, then exit to find plenty of enemies to stop you from getting away. One of those enemies will be a Heavy, which will probably be the first one you see as soon as you exit. Kill them all and bring the diamonds to the drop-off point to finish the mission.