In Resident Evil 6, you will earn certain medals at the end of a chapter or mission for completing certain tasks. There are a total of 264 different medals across Campaign, Agent Hunt and Mercenaries, so earning 150 shouldn't be too difficult. 175 medals are found in the Campaigns, so you technically don't even need to touch the other modes in order to earn this, though chances are you will only earn roughly 120 medals through Campaign during your first playthrough. If so, then play some Agent Hunt and Mercenaries, trying to complete as many different medals as possible.
If you want to check how many total medals you have, then go to Dog Tags and read the numbers under the three respective dog tags: Decorated Agent, Decorated Mercenary and Decorated Hunter. Add these three numbers for your total.
For a list of the available medals and how to earn them, please see this Medal List, courtesy of Sixation.