You don't have to calculate the position of or rotate the camera. You just have to child it, as the same way you did with the CharacterController. Make sure, you have made that through the code.

var camera : GameObject;
var platform : GameObject;
function Update(){
camera.transform.parent = platform.transform;
}

Do not move the camera inside a FixedUpdate. There is no need for that. The pruprose of a camera is to shoot the scene, not playing physics.

Therefore, it is a good idea to fix the camera position juste before the scene is rendered.

Fortunately Unity provides juste the right event for that OnPreRender (edit: not the best solution, see below):

void OnPreRender()
{
    transformFrom = _cameraTransform.position;
    transformTo = player.position + player.TransformDirection( 0,4,-8 );

    _cameraTransform.position = Vector3.Lerp( transformFrom, transformTo, Time.deltaTime * cameraMovementDamping );

    _cameraTransform.LookAt(player.transform.position);
}

If this doesn't work (I did not try it myself), you can still update the camera position in a latter phase using LateUpdate:

void LateUpdate()
{
    transformFrom = _cameraTransform.position;
    transformTo = player.position + player.TransformDirection( 0,4,-8 );

    _cameraTransform.position = Vector3.Lerp( transformFrom, transformTo, Time.deltaTime * cameraMovementDamping );

    _cameraTransform.LookAt(player.transform.position);
}

edit: LateUpdate is the method to use. OnPreRender does not change position/rotation but can be used to set/unset effects such as fog.