You have to beat different trainers to trigger the deck unlocks.
In the training tournament cups, when selecting who to play against, you will notice at the bottom there is a tournament ladder thingy. Each circle represents a trainer to beat. Once you have beaten the trainer, a green checkmark will appear.
The ladder progress is specific to the current deck you are using. Meaning, the green checkmarks only indicate who you have beaten with the current deck only, so switching to another deck will show you a completely different ladder.
To unlock cards in the current deck you are using, you want to face as many different trainers that you have not beaten before and get those checkmarks.
Note: You can directly click on a circle in the ladder to select the corresponding trainer. And aside from this method, I don't think there is any other way to unlock cards.