Brock won Chansey while she was an egg in a contest in All Dressed Up With Somewhere To Go!.
The egg hatched revealing the Pokémon to be a Happiny at one of the Pokémon Centers. As a Happiny, she possessed enormous strength, using that strength to throw Brock into the air when she hatched. Later in the series, Happiny lifted an enormous boulder with one hand and also lifted Ash's Grotle and carried it around Canalave City when it got tired.
While on the boat to Twinleaf Town, Brock's Happiny evolved into Chansey while trying to help a sick Pichu. It is currently unknown if Chansey has kept the strength she had as a Happiny but she knows the move Soft-Boiled which she used to heal Pichu on the boat.