Suicune first appeared at a lake in Johto, where a Goldeen was feeling ill due to dirty water. Suicune used its power to restore the lake water Goldeen was in and heal it. Later, a group of Pokémon hunters started to deliberately pollute the lake in order to force Suicune to purify it again and again. By the time Ash and Goh arrived, Suicune's repeated use of its purification power had left it low on stamina, allowing the hunters to seize it with nets and ropes. Desperate to save Suicune, Goh threw a Poké Ball at it and caught it.
Leaving Ash to battle the hunters, Goh ran away to get Suicune to a Pokémon Center, but tripped and ended up dropping the Poké Ball, allowing Suicune to get out. It climbed to a nearby tree to recover, so Goh started caring for it himself, treating its wounds and feeding it Berries to restore its stamina. When the hunters showed up again, Suicune witnessed Goh's selfless actions to protect it, and it agreed to fight by his side, quickly defeating the hunters and freezing them in a block of ice, allowing Officer Jenny to arrest them.
With the hunters apprehended, Goh offered to release Suicune so that it could continue its journey, but it chose to remain his Pokémon. As such, Goh then brought it to Cerise Laboratory and showed it to Professor Cerise, Ren, and Chrysa. Soon after, though, Suicune departed to continue roaming the world, which Goh was perfectly happy with, as he felt that Suicune was still present in spirit.
In Big Brother to the Rescue!, Suicune temporarily returned to Cerise Park, where it helped to purify the water polluted by Goh's Grimer and restore order among the other Pokémon before departing once more. Before leaving again, Suicune acknowledged Yamper's efforts to keep Cerise Park in order.
Suicune is a kind and caring Pokémon. Like all of its species, it has the power to purify dirty or polluted water, and is very dedicated to using this power, as seen when a group of Pokémon hunters exploited this sense of duty to force Suicune use its power to the point of exhaustion. Being a Legendary Pokémon, it's also very powerful.
Initially, Suicune didn't take kindly to Goh catching it, but upon seeing his selflessness in how he healed and protected it, it started to view him as worthy to be its Trainer, even allowing him to command it in battle. Despite this, however, it has not given up on roaming the world to continue its task of purifying waters. It may still show up at Cerise Park on occasion to watch over the Pokémon residing there, as shown by it having witnessed Yamper's work at the Park.