Add the following to your chosen theme's template.php file - and assuming you've created an additional field in your user records called "field_profile_realname"...
function [yourthemname]_preprocess_page(&$variables) {
$arg = arg();
if ($arg[0] == 'user' && is_numeric($arg[1]) && empty($arg[2])) {
# we're looking at a user profile page
$user_data = user_load($arg[1]);
if (isset($user_data->field_profile_realname['und']['0']['value'])) {
$variables['title'] = $user_data->field_profile_realname['und']['0']['value'];
}
}
}


