Connect to your web server using an FTP program. Create a new folder on your server, e.g. /counter/.
Upload the whole script including all files and folders to your server. Make sure the original file structure appears on the server.
All text files have to be uploaded in text mode (ASCII mode) and all binary files (like images or PDF files) have to be uploaded in binary mode. The simplest would be to set the mode to "auto". This way your FTP program takes care about the correct transfer mode.
Please make sure that your FTP program preserves the file names and doesn't lower case them.
Make sure the folder /logfiles/ is writable by the script (chmod 777).
Most, if not all, FTP programs allow you to change the file permissions. That can either be done with a command line and the command chmod or with some clicks on menus and checkboxes.
Here is a guide with examples of common FTP programs: