Check if specific username is logged in

wp_get_current_user then compare the object that’s returned user_login proprety with the specific username you’d like to check against.

<?php
add_action('admin_init', 'wpse74389_check_username');
function wpse74389_check_username()
{
    $user = wp_get_current_user();

    if($user && isset($user->user_login) && 'username_to_check' == $user->user_login) {
        // do stuff
    }
}