Install PHP5-FPM as described here.

Install Nginx:

apt-get install nginx

Check for the correct path to the socket used by php5-fpm in /etc/php5/fpm/pool.d/www.conf (to be used in the nginx configuration):

listen = /var/run/php5-fpm.sock

Create a virtual host in /etc/nginx/sites-available/:

server {
    listen  80;
    root /var/www;
    index index.php index.html index.htm;
    server_name YOUR_SERVER_NAME;
    location / {
        try_files $uri $uri/ /index.html /index.php;
    location ~ \.php$ {
        fastcgi_split_path_info ^(.+\.php)(/.+)$;
        # NOTE: You should have "cgi.fix_pathinfo = 0;" in php.ini
        fastcgi_pass unix:/var/run/php5-fpm.sock;
        fastcgi_index index.php;
        include fastcgi_params;

Replace root, server name and the above checked socket to fit your environment. In this case I call the file your_new_vhost.

Activate the new virtual host by placing a symlink in /etc/nginx/sites-enabled/:

ln -sf /etc/nginx/sites-available/your_new_vhost /etc/nginx/sites-enabled/your_new_vhost

Restart nginx: /etc/init.d/nginx restart