To take part in the Rank D Ninja World Tournament, start the "Ninja World Tournament" from the main menu and select the option to continue the game. You will be asked to select your character again, after which you will land on the beach again. During the cutscene you will learn that you have gained access to another part of the island (Rank C - located in the caves), and that you've unlocked the access to Challenges. To take part in them, go back to D Rank Tournament Reception - a nearby window will now be hosting an NPC with a list of challenges.
Rank C of Ninja World Tournament will be unlocked after you've finished the previous rank - D. The gameplay is fairly the same - you still have to play a qualifying round, after which there's a preliminary and finally main round, after which you will complete the C Rank of the tournament and gain access to the next one (B) and further section of the island.
To start your journey in this class, go to the previously mentioned cave, in which the events of Rank C of tournament take place. The entrance to the cave can be found to the North-West from the place you landed on the beach. You can also use the quick-travel option, provided by an NPC called Teleportation Ninja. It's a special NPC which appeared after you've completed Rank D of the tournament - with that character you can select a place to which you want to teleport in a matter of seconds. As you unlock higher ranks and further parts of the island, the Teleportation Ninja will give you access to wider range of teleports. Teleportation Ninja can always be found near a Reception window.
After entering the cave, run along the tunnel, until you reach the plaza, where all the shops and Employment Office are located. You can, before you start the tournament, do some side-quests, if you want to. If you want to unlock new allies for the ninja tournament, go to the "Ninja World Tournament - Ally Events" chapter of this guide, to "Basking in the Limelight" mission, and if you want to do some extra work, go to Employment Office.
If you want to enter the Rank C ninja tournament, go to C Rank Tournament Reception, to register yourself, after which you will take part in the qualifying round.