Find the Tomb
Reach the marked area, where there will be Buck waiting for you. After you have talked to him, jump down the cliff into water. Get out of water on the left/western bank and climb the vine shown in the screenshot. This will take you to the cave that you need to enter.
After a moment's stroll, you will notice an ancient door to your left. Pass through the door and go along the tunnel, until you reach a wall with the vine hanging down from it. On the top, you will notice a pirate through the hole in the ground. If you have the Death from Above skill available jumping onto the enemy will do the trick.
Below, there are two other pirates. The Chained Execution skill will prove very useful here. The area further on is being patrolled by a dog and an armored enemy. I recommend that you start an open fight here.
After you dispose of all of your enemies, enter the area next to the zip line (the screenshot) and collect the flak jacket from the crate.
At the end of the zip line, there will be, most probably, a pirate waiting for you. To kill him, you need to jump onto him. Now, you will be up for another fight. Find a cover and kill all of the enemies.
Get to the very bottom of the caves, by means of zip lines. After you zip for the last time, you will notice a torrent of acid to your right. Get down there on the wooden platforms shown in the screenshot. Down below, there are further pirates.
Keep to the right of the cave as you descend. You will, eventually, reach an area, in which acid will be squirting from between rocks (the screenshot). You need to pass when all of acid falls down onto the ground. That is where you will encounter another group of enemies.
Reach the end of the cave and make sure that you do not step into acid on your way. You will encounter further enemies there. After you kill every last of them, climb up to reach the stairs that the enemies used to stand on, using the vines on the wall (the screenshot). Once on the top, climb the stairs to find another ancient door, which you can open using the compass.
Find the knife
Reach the end of the path and jump onto the wall where there are vines hanging down. Jump down and go right over the wooden scaffolding. When you reach the very bottom, watch out for the sulfur fumes shown in the screenshot. Move over the surrounding elements of the environment that are elevated a bit, after the fumes fall for some time.
Eventually, you will reach vines that you need to use to climb higher. Then, jump onto the ledge and go towards the stairs . Use the compass on the statue shown in the screenshot. This way, you will open a descent. As you go down, keep killing monitors.
Once you collect the knife in the tomb, start running towards the marked exit. Crouch, enter underneath the lowering ceiling and start pressing the correct keys to crawl over to the other side. This way, you will leave the tomb and complete this quest.