There are 5 properties to a Chao
Swim
Fly
Run
Power
Stamina
Raise all these to Level 10 for the achievement.
You will be able to raise them using the "Tubes" dropped by enemies or collected Animals. The Tubes are colour coordinated and will raise the stat according to that colour where as the Animals can do multiple stats at once but can cause progress between levels to decrease. Example: A Penguin increases progress for Swim but also decreases progress of Power. Dont worry about that too much as it will never reduce your level.
Tube Colours are as follows:
Red - Power
Green - Run
Yellow - Swim
Purple - Fly
To raise Stamina, Feed you Chao the Round Fruit that grows in the garden.