Update cart lineItems through ajax
There are two major approaches.


The easier approach is to create a partial template that displays the cart lineItems, and simply ajax load that template in to a div that is your cart display after you've done the lineItem update.  However, this has the disadvantage of making the whole cart section update in a visible way, which look a lot less finessed than the second approach.
In  short, you'll need to use the cart object in the json response to re-draw just the appropriate parts of your cart display.


Here's an example of approach 2, (don't focus on the specifics below which are from a live site - but rather this should be enough to give you the idea) - if the user e.g. adjusts line item quantities, we grab the appropriate parts of response.cart and then do some local calculations and update the appropriate parts of the page:

            $.post('/actions/' + $('input[name=action]').val(), form.serialize(), function(response) {

            // Change in qty successful
            if (response.success) {
                if (response.cart.lineItems[id]) {
                    var subtotalNew = parseFloat(response.cart.lineItems[id].price * response.cart.lineItems[id].qty);
                    utilities.replacePrice($subtotal, subtotalNew);
                }

                navCart.update(response.cart);
                cart.update(response.cart, true);
            }


...where, for example, utilities.replacePrice is:

replacePrice: function($obj, price, discount) {
    var text = module.exports.currency(price);

    // Handle discounts
    if (discount) {
        // Supplied as negative number - because we want it formatted differently, we make it positive
        var discountPrice = module.exports.currency(price * -1);
        text = '- (' + discountPrice + ')';
    }

    $obj.fadeOut("fast", function(){
        $obj.text( text );
        $obj.fadeIn("fast");
    });
},


...and the cart.update functions are similar and take care of updating the nav bar cart and the totals and shipping sections.

I will warn you that implementing in full a completley ajax updating cart page is a fair bit of work and complexity to get right, and the actual implementation will depend greatly on your checkout flow, which you should have thoroughly worked out BEFORE you implement this, or you can quickly tie yourself in JS knots.

If you want to see a fairly completely ajaxed Commerce cart in action, add this to your cart and then visit your cart and muck about changing quantities and removing things etc:
https://imagescience.com.au/products/inkjet-print-retouching-kit