Once you reach level 7, the bounty board will become available to you at the inn. It is located opposite the forge. You can select up to three different contracts at a time. Make note of the specifics of the contract such as difficulty and level.
Choose the level and difficulty required, and then complete the task it asks of you. I believe you will need to finish the level in the same run for it to count as completed.
The achievement should pop at the end of the level, so you don't even need to cash in on your reward (but why wouldn't you anyway). A quick note, both myself and a friend of mine unlocked this achievement on our second completed contract. I am unsure as to why this is, but just keep that in mind in case it doesn't pop on your first completion.