Step 1:
In arcade home you'll see the following:
Click it!
Step 2:
The link will direct you to the active tournaments screen. You'll find the tournament you want to join there. Click "Register!" on the tournament you want to join.
Step 3:
You've now joined the tournament! Now you have to wait for the others to join. When everyone has joined, you'll see "1 running tournaments" in the arcade home... Click it.
It will direct you to the tournament screen, pretty much like in Step 2. There is also a link to play. After you've played the game it directs you to the normal submit score screen. Just submit the score and it will be updated in the tournament database.
Enjoy playing and remember fair play!