It's actually quite simple. To become king, you simply acquire all five of the Royal Artifacts.
How do you do that? Well, every so often, one of the royal artifacts will be mailed to whichever octopus is winning each of the contests. The five contests are:
See individual pages for the scoring of each test.
If your octopus is winning a contest, and therefore receives an artifact, but then starts losing, the judges will instruct you to mail that artifact to the new winner. It is your responsibility to do so promptly. Don't worry, aritifacts should arrive with little mailing containers. If you do not receive an artifact in a timely manner, the judges will provide a substitute, and penalize the miscreant appropriately.
Good luck!