This achievement requires that you complete all the challenges at a 1-star grade.
The challenges can be located right off the Main Menu > Challenges. There are four distinctly different challenges, designed around teaching you/testing your knowledge of the basics of Brink. They are a great place to start when picking up the game, and of course they are the only way to unlock most of the weapon attachments.
These challenges must be done Solo to earn the achievements ('King Of The World!' is earned by achieving a 3-star grade on each of them) and achieving your second and third stars is not a case of completing the courses quickly, but repeatedly.
Each course has three iterations, more difficult than the last. Fortunately, the first tiers are almost insultingly easy, with the free-running challenge offering the only chance of failiure - simply because finding the fastest route is trial and error.
I strongly recommend completing each course to its third tier before moving on, as practice makes perfect, and the difficulty curve will feel much more natural in one smooth procession than if you were to come back to it.
If you're having trouble at all, check the guide for 'King Of The World!' because TA member Maka91 has compiled a fantastic set of video guides that will surely help you master these challenges.