How do you play the database puzzles?

Simply click on the tab on the bottom part of the screen titled "Internet". The first set of computers that you are "supposed" to hack is the messenger database. Although it isn't right to call them as database puzzles as the concluding puzzles are simulating different hacking obstacles such as the Town Archive, Serial Killer Folder, and what not.

Otherwise, If you are referring to the actual way how to play them (technical details), simply click on the link on the "How to Play the Puzzle Games", found on the Game Basics tab on the complete Game Guide for a complete explanation. Hope this helps!