jQuery.ajax returns 400 Bad Request

Add this to your ajax call:

contentType: "application/json; charset=utf-8",
dataType: "json"

I think you just need to add 2 more options (contentType and dataType):

$('#my_get_related_keywords').click(function() {

    $.ajax({
            type: "POST",
            url: "HERE PUT THE PATH OF YOUR SERVICE OR PAGE",
            data: '{"HERE YOU CAN PUT DATA TO PASS AT THE SERVICE"}',
            contentType: "application/json; charset=utf-8", // this
            dataType: "json", // and this
            success: function (msg) {
               //do something
            },
            error: function (errormessage) {
                //do something else
            }
        });
}