Create a Place on Your Remote Host to Install "Make Contest"
If this is a new site, you will probably want to have your site in home directory. If you have an existing site and want to add "Make Contest", you might want to add a subfolder / subdomain to hold your "Make Contest" site.
Copy Files and Directories of Your Local "Make Contest" installation to the Remote Host
Upload all files and folders of your current installation using file manager in your Control Panel of FTP-client of your choice.
Copy the Contents of Your Local MySQL Database to the Host MySQL Database
"Make Contest" keeps the info about all the contents of your site in the MySql database. You need to copy this information to the host database. To do so you need to export data from you local database and then import it into your host database. This can be done with phpMyAdmin; if you are not sure how to export / import database please search online tutorials to help you on the subject.
Configure the site to work on remote server
Manually edit your cfg.php file with remote host info: server_name, site_url, upload_full_dir, dbhost, dbname, dbuser, dbpasswd, etc.
At this point, your "Make Contest" site on your host should be operational with the same information as your localhost site.