Enter the room with the chains and climb the chain on the right.
Climb until the spherical metal weight prevents you from going any higher.
If you're not used to this style of camera control, it may help to turn Lara so she's facing the other chain, as shown above. Then move the camera directly behind her by pressing the center-camera/display HUD button (J/L2/LB). Adjust Lara's angle if necessary until she's squarely facing the other chain. Then jump forward and she should grab it.
NOTE: It really doesn't matter which way Lara is facing when you jump as long as the camera is pointed where you want her to go. But centering the camera behind her makes it easier to aim for the chain.
Once Lara is hanging from the second chain it begins to descend. So climb quickly. As the on-screen hint suggests, you can tap Interact in time with Lara's movements to climb faster.
As soon as Lara is higher than the round weight on the other chain, jump to grab onto that chain above the weight. Again, you can turn Lara toward the chain if you like, but it doesn't matter which way she is facing. Just point the camera directly toward the first chain and jump forward and she should catch it.
If you don't climb fast enough, the oblong weight on the chain Lara is holding will prevent her from climbing high enough to clear the round weight on the other chain. If that happens, drop down and try the whole thing again.
When you have grabbed onto the first chain above the round weight, you can take your time. Climb up so Lara is level with or just above the raised opening in the wall. Point the camera toward that opening and jump forward to land there.
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