Unlocking the Silver Trench
Follow the storyline of your version of "Pokemon Mystery Dungeon" to the point where you can unlock the Silver Trench level. You must recruit Latias and Latios, as well as befriend the legendary trio of bird Pokemon: Articuno, Moltres and Zapdos. In addition, you must complete the Stormy Sea and Pitfall Valley levels to be able to enter the Silver Trench.
Legendary Island's Secret
Go to Legendary Island and talk to one of the legendary birds. Articuno, Zapdos and Moltres will give you the Whirl Symbol to open the Silver Trench dungeon. You will need to bring a Pokemon that knows the move Dive to enter Lugia's home. Stand in the middle of the dark blue water on Legendary Island and dive into it to visit the Silver Trench.
The Silver Trench
Explore the Silver Trench and fight your way past the local Pokemon to reach level 99. Some of the many Pokemon living in the Silver Trench include Slowking, Muk, Shellder and Cloyster. More dangerous Pokemon lurk in the higher levels of the Silver Trench, such as Cradily and Lanturn. Bring along Electric- and Grass-type Pokemon to combat the mob of Water-type Pokemon on the way to Lugia's chamber.
Facing Lugia
Walk up to Lugia on the 99th floor of the Silver Trench and engage it in a Pokemon battle. Although all of Lugia's stats are on the higher end of the scale, it has an inherent weakness to Electric-, Ghost-, Rock- and Dark-type attacks. Lugia will join your Pokemon Team upon defeat.