When you have finished Act V in Campaign Mode, you will unlock Adventure Mode for all your characters. This is a special mode that forgoes the normal quests, in favour of Bounties and Nephalem Rifts.
When you first start Adventure Mode, you will see Tyrael to the left of the Waypoint. Talk to him and he'll explain what Bounties are and how to access them. When you're done, open up your map and the Waypoints with Exclamation Marks over them will mean there is a bounty available.
Travel to the Waypoint and fulfill the requirements. Do this another four times and the achievement will pop. You can do this with a high level character on Normal difficulty to make it go faster.