The problem you have is here:
movement.x = Input.GetAxis("Horizontal") * speed;
movement.z = Input.GetAxis("Vertical") * speed;
You're applying movement in "world-space". If you want to move in relation to your character's orientation, you need to use the direction vectors on your character's transform:
movement = (transform.right * Input.GetAxis("Horizontal")) + (transform.forward * Input.GetAxis("Vertical"));
movement *= speed;
So instead of moving along "x" or "z" you're moving along "transform.right" or "transform.forward". Depending on your character's orientation you may need to use other transform vectors. Check out the "up, forward and right" fields in the transform documentation here: http://docs.unity3d.com/Documentation/ScriptReference/Transform.html
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