From the 'Windmill' base camp, the entrance is to the east. In the first circular chamber, search the left side to find a [GPS CACHE] among the grass. Keep moving until Lara slides down the slippery slope, and be ready to get a good grip. Move over the beams until you reach an intersection with some [SALVAGE]. The left path leads back out of the cave, the right path moves you deeper into the cave, leading to the puzzle area of the tomb.
The puzzle here is extremely simple. First jump from the ramp on the left side of the area to break the platform that has a jerrycan on it. The rubber wheel on the right side can be swung against the nearby platform to the right. You should already have three jerrycans by now. Place these at the end of the ramp on the right side of the area so that it stays down on the smaller side (you'll first need to step onto where you dumped the jerrycans.) Now you can run over it and quickly swing your axe into the climbable rock wall to reach the chest on the left side; climb over the beams to reach it.
Chamber of Judgment Tomb puzzle solution.
Your reward is, as always, [1250 XP], [250 SALVAGE] and the [SHANTYTOWN GPS CACHE MAP]. You'll also receive [HANDGUN PARTS 2/3] here. Exit the cave entirely.
Move on top of one of the crates and ride it to the other side. A big and tough firefight starts now; especially on the hard difficulty things can get hectic. Equip your rifle and quickly dispatch the guys sliding down to your platform. After taking care of those, start dispatching guys in the tower up ahead. ONLY shoot a red explosive barrel if you're 100% sure someone is standing closeby, otherwise it's a waste of time and of the barrel. Keep moving from cover position to cover position as there are a lot of pyro's throwing molotovs and dynamite at you. If enemies reach your platform, back away and hide behind the larger pillars in the corners with your shotgun equipped. If you've taken care of those, quickly aim some precision shots at the guys in the tower. It's possible to destroy some of their cover positions with burning arrows, but this is only worth the trouble when only a few of them remain. [400 XP].
If I've kept track of everything correctly, you can find [SHOTGUN SHELLS], [ARROWS], [ARROWS], [RIFLE AMMO], [ARROWS], [ARROWS], [RIFLE AMMO], [SHOTGUN SHELLS], [ARROWS], and [RIFLE AMMO x2] around here.
Use the rope to reach the tower, climb to the top and make your way to the rock wall on the southeast (you'll have to get a good grip after the scene). [100 XP]
Proceed to find the 'Vista Tower' day camp. Continue over the large bridge, head inside the house up ahead and make your way to the bridge. Leap over and follow Roth's instructions closely and you'll make it across. When he tells you there are too many, quickly jump on the nearby roof and run down the collapsing walkway until a rock wall comes up. Climb it [100 XP], then move through the tunnel. [100 XP]. Move further inside the linear cave and you'll get even more experience. [100 XP].