You will unlock bounties after completing Act V, then you will be able to access Adventure Mode. Go Tyrael, he will explain how bounties work.
To complete the 5 required, you need to go to any of the Waypoints, you will see gold exclamation points on certain locations, go to these areas and complete the task (defeating enemies or bosses etc.)
To make these easier and faster, put the game on Normal Mode, and use a high levelled character!