JSON Object Value Show Undefine

You either need to tell jQuery that you expect a json response from the server:

$("#txt-cmt").keypress(function(e) {
  if (e.which == 13) {
    var comment = $(this).val();
    var message = {
      action: 'show_comment',
      user_message: comment
    };
    $.post(ajaxUrl.url, message, function(data) {
      console.log(data.user_name);
    }, 'json');
  }
});

or you need to parse the response as json with JSON.parse().