Uncaught TypeError: Cannot read property 'ajax' of undefined

Did you try doing what the rest of the code is doing, using jQuery

jQuery.ajax({
  URL: "/AdminPanel/News/DeleteNews",
  data: { "newsId": 1 },
  dataType: "json",
  type: "POST",
  success: function (msg) {
  alert(msg);
}

You can wrap your code in a DOM ready function that sets the value of $ locally in the function scope, that way you can always use $

jQuery(function($) {
    // code goes here
});