Make character follow mouse using Character Controller

Question

I have a 2D space shooter game (imagine something like Chicken Invaders) that has been released on iOS and Android. I am now trying to port it to WebGL, and to do that, I'm trying to implement movement using the mouse. Having previously had success in implementing the movement using Unity's built-in character controller on the mobile platforms, I'd like to keep using it in the WebGL build to keep the code-base small and with as little platform-specific implementation as possible.

Therefore, what approach should I use to get the ship to follow the mouse cursor?

PS: On mobile I used touch.deltaPosition with characterController.Move() to have the ship follow the finger's movement.

EDIT: Current method

Input manager

public Vector2 GetInput () {
        switch(type) {
            case PlatformType.Computer:
                if (Input.GetButtonDown("Jump") || Input.GetAxis("Fire") != 0f) {
                    thisDelegate.InputDidReceiveFireCommand();
                }
                if (!Input.mousePresent) {
                    temp.x = Input.GetAxis("Horizontal");
                    temp.y = Input.GetAxis("Vertical");    
                } else {
                    if (Input.GetAxis("Mouse X") != 0f || Input.GetAxis("Mouse Y") != 0f) {
                        temp.x = Input.GetAxis("Mouse X");
                        temp.y = Input.GetAxis("Mouse Y");
                    } else {
                        temp.x = Input.GetAxis("Horizontal");
                        temp.y = Input.GetAxis("Vertical");        
                    }
                }
}

PlayerController

void HandleInput()
    {
        mvm = InputController.GetInput();
        if (GetComponent<InputManager>().type == InputManager.PlatformType.Mobile) {
            mvm *= Time.deltaTime * initialVeloc / 10;
        } else {
            if (mvm != Vector3.zero) {
                mvm -= transform.position;
            }
        }
    }

void Update () {
    characterController.Move((Vector2)mvm);
}

Current method makes the character jitter next to the center of the screen.

Answer 1

I've modified my script and I'm now taking the absolute mouse position instead of the delta. Here's how it looks now:

Input Manager:

            if (Input.GetAxis("Mouse X") != 0f || Input.GetAxis("Mouse Y") != 0f) {
                temp = Input.mousePosition;
                temp.z = 12.5f;
                //                Vector3 initialMov = mvm;
                temp = Camera.main.ScreenToWorldPoint(temp);
                temp -= transform.position;
            }

Controller

mvm = InputController.GetInput();
mvm *= Time.deltaTime * initialVeloc;
characterController.Move((Vector2)mvm);

This method works just as intended and seems more efficient than the RayCasting method, in terms of CPU usage.