How do I get evidence of Mercer Frey's betrayal in Skyrim?

Presumably, you're in the quest "The Pursuit". In your Quests tab, set this quest as a priority, which will give you a compass for direction. Once you break into Mercer's house, the compass will guide you to a room behind a "Suspicious Cupboard/Dresser". You will then steal Mercer's personal notes and then go back to the Thieves Guild, where you will meet with Karliah and Brynjolf. These notes prove Mercer's betrayal. They will go through some dialogue, the rest of the guild will turn against Mercer, and a new quest to become a Nightingale (after which is a quest to defeat Mercer) will start.