If I understand correctly, you want to just use Drush on remote site. You don't need to do anything with aliases.drushrc.php for it. You need to have access to SSH on your remote server, there are many tutorials on the net how to use SSH. You need to upload Drush through SSH on server - create a folder, like "programs" (it would be nice if it would be not accessible through http) and put there drush-7.x-5.8.tar.gz . Untar it with command:
tar -zxvf drush-7.x-5.8.tar.gz
Edit your .bash_profile file located in your home directory or create it (you can use command "nano .bash_profile" in your home directory), and put there:
export DRUSH_PHP=[absolute path to your php]/bin/php-cli
For example: "export DRUSH_PHP=/usr/local/php54/bin/php-cli" - you can check this path in output of your phpinfo().
Also add this line:
After this if you are logged on your remote server through SSH, you can go to directory of your Drupal site and use command "drush status" to check if everything works ok. Voila - you have drush installed on your server.