Things To Remember

First, we need a camera and put it inside the player so the camera can move with the player and move in the player's direction. Now add the script.

Code

// VARIABLES
[SerializeField] private float mouseSensitivity;
[SerializeField] private float xRotation;
[SerializeField] private Transform player;

// REFFFERNCE
//////////////////////
// RUN ONCE AT START
void Start()
{
    // Make sure mouse is locked in the game.
    Cursor.lockState = CursorLockMode.Locked;

    // Get player body.
    player = transform.parent;
}

void Update()
{
    // VERIABLE
    float mouseX = Input.GetAxis("Mouse X") * mouseSensitivity * Time.deltaTime;
    float mouseY = Input.GetAxis("Mouse Y") * mouseSensitivity * Time.deltaTime;

    // Getting where to move mouse.
    xRotation -= mouseY;
    // Stop mouse rotation to get beyond one point.
    xRotation = Mathf.Clamp(xRotation, -90, 90);

    // Rotating player for rotating side by side.
    transform.localRotation = Quaternion.Euler(xRotation, 0, 0);

    // Rotating camera up and down.
    player.Rotate(Vector3.up * mouseX);

}

Note: Make sure that when you set up your camera, you get the X rotation value of the camera and put it in the xRotation variable either in the inspector or in code by giving [SerializeField] private float xRotation = 15; (I have taken 15 as an example).

Thank You