Bounties can be acquired a number of ways but the most organic way is to make sure you get the killing blow on an enemy with a bounty on their head. The game might make it seem like the only way to get credit for this achievement is to do bounty missions in the space port, however any craft you take down that has a bounty on it counts towards this achievement.
Anytime you start a space battle, deploy your scanner with the
button and then selecting the appropriate icon in the radial menu. Your on-board computer will say "Bounty Detected!". Simply hover your cursor over every ship until you locate the ships with bounties, mark them and destroy them.