Off the rules-page for DTs:
"Every day we organize 12 Tournaments in which you can win Clintz and Credits.
If you wish to take part, all you need to do is play during the Tournament period in one of the Tournament rooms with a Deck that is valid for that room: Type I, Type II, ELO, etc. (Game modes and Deck formats)
The rooms valid for Daily Tournaments are easy to spot as they show this icon:" [a golden trophy-icon]
You don't need to signup. Just go ahead and play and as soon as you scored your first point, you're a participant.
Bacially, all you have to do is join either Type 1 Fights, or Type 2 fights during the tournament times. I believe you are GMT so they start on all the odd hours of the 24 hour clock. They last for a hour and its advised you play for the full time to have a good chance of winning good clintz. You get 50 clintz for just entering, and 1 credit if your in the top 3rd. Prizes increace when you reach the top 100.