First, you must complete the smaller tournaments before unlocking the big tournaments. Once you have unlocked the larger challenges and completed the last of them, the trophy will unlock. There are several ways to complete each large tournament:
1) Solo. Grind it out for hours fighting enemies equal to your level.
2) Co-op. Have your main character be a low level (10-20) and your second a high level (50+). Have the low level character start up the larger tournament(s) you still need to complete and use your high level character to kill enemies. This is the safer method compared to playing solo.
3) Online. Jump in a game that is playing in the tournament(s) you need to still complete. You only need to be present for the last wave of the last round in order to unlock the trophy.
Happy Hunting :)