1) Fly to Cianwood City.
2) Go north of the Pokemon Center and talk to Eusine. He will challenge you to a battle, which you must win to begin the quest. (He will use Drowzee, Electrode, and Haunter.)
3) Fly to Ecruteak City and go East to Route 42. Surf across the first body of water and continue east where you will find Eusine a little southeast of the staircase.
4) Talk to Eusine and he will give you the sidequest “Legendary Suicune” which will direct you to go to the Dragon’s Shrine in Blackthorn City.
5) You will need Surf and Whirlpool going forward. Once you enter the Dragon’s Shrine, your quest will update to read lines on the pillars. Stand to the right of the 2 left pillars and interact with them. Stand to the left of the 2 right pillars and interact with them.
6) Time to find Eusine in Mt. Mortar. (Go back to where you talked to him earlier on Route 42 and go up the staircase and enter Mt. Mortar.
7) Use the map above to locate Eusine inside of Mt. Mortar.
8) Interact with Eusine and he will challenge you to another battle. His team will consist of Electrode, Gengar, Gengar, Hypno, and Hypno.
9) After defeating Eusine, Suicune will spawn beside you and you can now battle him.