You can do this by inserting the following code within the side bar div or sidebar menu. Edit the code to match up with your feedburner account.

<?php
$url = "https://feedburner.google.com/api/awareness/1.0/GetFeedData?uri=yourURIhere";
$ch = curl_init();
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch, CURLOPT_URL, $url);
$data = curl_exec($ch);
curl_close($ch);        
if ($data) {
    preg_match('/circulation="([0-9]+)"/',$data, $matches);
    if ($matches[1] != 0)
        $fb['count'] = $matches[1];
    $fb['lastcheck'] = mktime();
    update_option("feedburnersubscribecount",$fb);
}
echo $fb['count'].' Readers!';
?>

Found on: yoast.com