Pokemon Crystal expanded on the story concept of Pokemon and was the first game to tie the plot into a legendary Pokemon, namely Suicune. You don't get to catch Suicune first time you see it, but you do have a chance before battling the Elite Four.

Burned Tower

When you arrive in Ecruteak you will notice the gym is closed. Go to Burned Tower to meet the gym leader Morty and his friend Eusine, who are studying the legendary Pokemon. Make your way to the center hole and fall in to meet the legendary trio, Suicune, Entei, and Raikou. The latter two will run off, but Suicune will parade around you before it takes off. Eusine and Morty will follow you downstairs, say something about how Suicune chooses a trainer worthy of it, then leave. You can now fight Morty to get the badge for Surf and continue your quest for Suicune.

Cianwood City

You can do this part or the next however you wish. Since it follows the badge order I will be explaining it as if I did this part first. When you get to Cianwood City, there is a ledge north of the town where Suicune will be hiding. It will see you and jet off across the water. Mysticalman Eusine will follow and declare he wishes to defeat you to prove himself to Suicune.

Mystical Man Eusine Drowzee Haunter Electrode
Battle Type
Single Battle
Level 23 Level 23 Level 25
Dream Eater
Mean Look
Route 42

Next you will see Suicune on Route 42. It is hiding in a mini-forest blocked off by a Cut-able tree. Chop it down and Suicune will depart and be ready for your next encounter.

Tin Tower

After beating Pryce and kicking Team Rocket out of Goldenrod City, you will receive a Clear Bell as thanks. Take it to the Tin Tower, where you will be stopped on the way by the Wise Trio. Defeat them to prove your worth:

After that you will be allowed into the tower to fight Suicune in the foyer.