If it's not shown on the list, click "browse" and search for it.
Then click the last option "Add a Non-Steam Game to My Library".
Click the "Games" tab on the top-left side of your Steam Client.
This method is assuming you're not using the "Big Picture Mode". To use Steam Overlay in a Non-Steam game, you must add that game into your Games Library and launch it there. This should include games such as Minecraft and Overwatch, to throw a few popular names out there. While Steam Overlay is not guaranteed to work for all Non-Steam Games, most games should work.