Bleach does kill bed bugs but I would only use the bleach to kill the bed bugs by washing bedding and pillow cases in the washing machine.
But I wouldn't recommend spraying bleach solution on your carpets, bed or other furniture since the bleach could damage your carpet and furniture.
A good way to kill bed bugs on your carpet and bed and other furniture is by mixing some pesticide in a steam upholstery cleaner and then steam cleaning your carpet and upholstery.