If i'm rotating my player on the Y-Axis and move it like
transform.Translate(Vector3.forward * Time.deltaTime)
it will move along the Z-Axis of the character (as expected).
If I used CharacterController.Move it will always move in the same direction, independent from its rotation.
How can I make the player follow it's rotation?
The problem is that you are translating the character forward on a global level. This means forward will always be in the same direction. Instead, use transform.forward. That way, you are translating the player on the forward vector on the player.
Another way to do it is using transform.rotation * Vector3.forward. This will rotate the global forward by the rotation of the player, returning the same thing as transform.forward.