For those who are here for quick instructions

1) Run game, memorize PID
2) Run: "C:\Program Files (x86)\Steam\GameOverlayUI.exe" -pid XXXX -manuallyclearframes 0
3) Replace the XXXX with your PID

Complete walkthrough

1) Run Tomb Raider Underworld
2) Open Task Manager
3) Find "tru.exe" (Description = Tomb Raider: Underworld)
4) Memorize the PID
5) Find the path of your Steam installation (Usually C:\Program Files (x86)\Steam\)
6) Press Windows key + R
7) On the Run window, and using your Steam installation path, you gotta write like so:
"C:\Program Files (x86)\Steam\GameOverlayUI.exe" -pid XXXX -manuallyclearframes 0
8) Replace XXXX with the PID you've memorized from the Task Manager
9) Go back ingame and enjoy the Steam Overlay! You can now take screenshots as usual!

Note: On step 7, YOU MUST USE THE QUOTATION MARKS!
Note 2: This process needs to be done every time you boot the game.

PID is not visible anywhere, HELP!
Windows 7:
On Task Manager, go to view » Select columns.
Usually it's the first option: PID (Process identifier)

Windows 10:
You must be in the Task Manager's "Details" tab, not the normal "Processes" tab.
If it's not showing up (By default, it should), right click on the top of any column and then "Select columns" to be able to activate PID column (2nd Option)