Comment Submission Without Page Refresh. No Plugins Used.

add_action( 'wp_enqueue_scripts', 'misha_ajax_comments_scripts' );

function misha_ajax_comments_scripts() {
 
	// I think jQuery is already included in your theme, check it yourself
	wp_enqueue_script('jquery');
 
	// just register for now, we will enqueue it below
	wp_register_script( 'ajax_comment', get_stylesheet_directory_uri() . '/ajax-comment.js', array('jquery') );
 
	// let's pass ajaxurl here, you can do it directly in JavaScript but sometimes it can cause problems, so better is PHP
	wp_localize_script( 'ajax_comment', 'misha_ajax_comment_params', array(
		'ajaxurl' => site_url() . '/wp-admin/admin-ajax.php'
	) );
 
 	wp_enqueue_script( 'ajax_comment' );
}

/*
 * Let's begin with validation functions
 */
 jQuery.extend(jQuery.fn, {
	/*
	 * check if field value lenth more than 3 symbols ( for name and comment ) 
	 */
	validate: function () {
		if (jQuery(this).val().length < 3) {jQuery(this).addClass('error');return false} else {jQuery(this).removeClass('error');return true}
	},
	/*
	 * check if email is correct
	 * add to your CSS the styles of .error field, for example border-color:red;
	 */
	validateEmail: function () {
		var emailReg = /^([\w-\.]+@([\w-]+\.)+[\w-]{2,4})?$/,
		    emailToValidate = jQuery(this).val();
		if (!emailReg.test( emailToValidate ) || emailToValidate == "") {
			jQuery(this).addClass('error');return false
		} else {
			jQuery(this).removeClass('error');return true
		}
	},
});
 
jQuery(function($){

	/*
	 * On comment form submit
	 */
	$( '#commentform' ).submit(function(){
	
		// define some vars
		var button = $('#submit'), // submit button
		    respond = $('#respond'), // comment form container
		    commentlist = $('.comment-list'), // comment list container
		    cancelreplylink = $('#cancel-comment-reply-link');
			
		// if user is logged in, do not validate author and email fields
		if( $( '#author' ).length )
			$( '#author' ).validate();
		
		if( $( '#email' ).length )
			$( '#email' ).validateEmail();
			
		// validate comment in any case
		$( '#comment' ).validate();
		
		// if comment form isn't in process, submit it
		if ( !button.hasClass( 'loadingform' ) && !$( '#author' ).hasClass( 'error' ) && !$( '#email' ).hasClass( 'error' ) && !$( '#comment' ).hasClass( 'error' ) ){
			
			// ajax request
			$.ajax({
				type : 'POST',
				url : misha_ajax_comment_params.ajaxurl, // admin-ajax.php URL
				data: $(this).serialize() + '&action=ajaxcomments', // send form data + action parameter
				beforeSend: function(xhr){
					// what to do just after the form has been submitted
					button.addClass('loadingform').val('Loading...');
				},
				error: function (request, status, error) {
					if( status == 500 ){
						alert( 'Error while adding comment' );
					} else if( status == 'timeout' ){
						alert('Error: Server doesn\'t respond.');
					} else {
						// process WordPress errors
						var wpErrorHtml = request.responseText.split("<p>"),
							wpErrorStr = wpErrorHtml[1].split("</p>");
							
						alert( wpErrorStr[0] );
					}
				},
				success: function ( addedCommentHTML ) {
				
					// if this post already has comments
					if( commentlist.length > 0 ){
					
						// if in reply to another comment
						if( respond.parent().hasClass( 'comment' ) ){
						
							// if the other replies exist
							if( respond.parent().children( '.children' ).length ){	
								respond.parent().children( '.children' ).append( addedCommentHTML );
							} else {
								// if no replies, add <ol class="children">
								addedCommentHTML = '<ol class="children">' + addedCommentHTML + '</ol>';
								respond.parent().append( addedCommentHTML );
							}
							// close respond form
							cancelreplylink.trigger("click");
						} else {
							// simple comment
							commentlist.append( addedCommentHTML );
						}
					}else{
						// if no comments yet
						addedCommentHTML = '<ol class="comment-list">' + addedCommentHTML + '</ol>';
						respond.before( $(addedCommentHTML) );
					}
					// clear textarea field
					$('#comment').val('');
				},
				complete: function(){
					// what to do after a comment has been added
					button.removeClass( 'loadingform' ).val( 'Post Comment' );
				}
			});
		}
		return false;
	});
});

<?php
add_action( 'wp_ajax_ajaxcomments', 'misha_submit_ajax_comment' ); // wp_ajax_{action} for registered user
add_action( 'wp_ajax_nopriv_ajaxcomments', 'misha_submit_ajax_comment' ); // wp_ajax_nopriv_{action} for not registered users

function misha_submit_ajax_comment(){
	/*
	 * Wow, this cool function appeared in WordPress 4.4.0, before that my code was muuuuch mooore longer
	 *
	 * @since 4.4.0
	 */
	$comment = wp_handle_comment_submission( wp_unslash( $_POST ) );
	if ( is_wp_error( $comment ) ) {
		$error_data = intval( $comment->get_error_data() );
		if ( ! empty( $error_data ) ) {
			wp_die( '<p>' . $comment->get_error_message() . '</p>', __( 'Comment Submission Failure' ), array( 'response' => $error_data, 'back_link' => true ) );
		} else {
			wp_die( 'Unknown error' );
		}
	}
 
	/*
	 * Set Cookies
	 */
	$user = wp_get_current_user();
	do_action('set_comment_cookies', $comment, $user);
 
	/*
	 * If you do not like this loop, pass the comment depth from JavaScript code
	 */
	$comment_depth = 1;
	$comment_parent = $comment->comment_parent;
	while( $comment_parent ){
		$comment_depth++;
		$parent_comment = get_comment( $comment_parent );
		$comment_parent = $parent_comment->comment_parent;
	}
 
 	/*
 	 * Set the globals, so our comment functions below will work correctly
 	 */
	$GLOBALS['comment'] = $comment;
	$GLOBALS['comment_depth'] = $comment_depth;
	
	/*
	 * Here is the comment template, you can configure it for your website
	 * or you can try to find a ready function in your theme files
	 */
	$comment_html = '<li ' . comment_class('', null, null, false ) . ' id="comment-' . get_comment_ID() . '">
		<article class="comment-body" id="div-comment-' . get_comment_ID() . '">
			<footer class="comment-meta">
				<div class="comment-author vcard">
					' . get_avatar( $comment, 100 ) . '
					<b class="fn">' . get_comment_author_link() . '</b> <span class="says">says:</span>
				</div>
				<div class="comment-metadata">
					<a href="' . esc_url( get_comment_link( $comment->comment_ID ) ) . '">' . sprintf('%1$s at %2$s', get_comment_date(),  get_comment_time() ) . '</a>';
					
					if( $edit_link = get_edit_comment_link() )
						$comment_html .= '<span class="edit-link"><a class="comment-edit-link" href="' . $edit_link . '">Edit</a></span>';
					
				$comment_html .= '</div>';
				if ( $comment->comment_approved == '0' )
					$comment_html .= '<p class="comment-awaiting-moderation">Your comment is awaiting moderation.</p>';

			$comment_html .= '</footer>
			<div class="comment-content">' . apply_filters( 'comment_text', get_comment_text( $comment ), $comment ) . '</div>
		</article>
	</li>';
	echo $comment_html;

	die();
	
}