Valve has released a new Steam client beta, which among fixing some issues with the streaming capabilities has also added an official FPS counter into the Steam in-game overlay.
The counter can be activated by going to the Steam client settings and opting into the beta. Once that has been done you'll find the option to toggle the counter through the in-game settings of the client, and can be viewed through the normal shift+tab overlay while in a game.
Those wanting to peruse the full patch notes can find them below:
Reduced CPU usage when drawing animated images or videos
Fixed video playback performance regression on Mac OS X and Linux
Added FPS counter to Steam Overlay
Fixed reloading settings values in the Steam Overlay
Improved capture performance in D3D9 games
Improved audio/video synchronization
Improved automatically adjusting video encoding bit rate when a change in available upload bandwidth is detected
Fixed opening the broadcast watch page from client UI (friends list, invite chat message, etc.) in the Steam Overlay
Improved capture performance in OpenGL games when hardware support is enabled on machines with newer Nvidia GPUs
Added first time use and other UI to Big Picture mode
Fixed first friend invite to a broadcast sometimes getting dropped