In order to earn Tournament Points you need to fulfil Objectives that can be of the following types:
PvP. Here you’ll need to battle at any location that has Units belonging to other Lords or Ladies, including enemy Castles, Beacons, and League Fortresses. You will earn more Points if you attack the League you are paired with in the Round, but this is not essential for success.
PvE. To succeed here, you’ll have to defeat Balur’s Units at Battlegrounds and Stoneheart Trials. You can find these from the Eagle's Nest in your Castle, but please be aware that Saga Quest targets do not count towards this Tournament task.
Development. You can also Build and Upgrade your Castle or Discover and Upgrade Lost Arts to earn Tournament Points.