If you want to upgrade Joomla 1.5 to 2.5 then first make sure to upgrade it to Joomla 1.7 first.
You can do this by going to http://yourdomain.com/administrator/index.php?option=com_joomlaupdate from your admin panel.
It will update it to nearest Joomla version.
After upgrading to this please take a backup through Akeeba backup or you can take your database and files backup separately.
Now just overwrite the existing files in the server with the new joomla 2.5 package (except the configuration file and installation folder). Please do not upload installation folder.
When you are done then again go to this URL http://yourdomain.com/administrator/index.php?option=com_joomlaupdate and then run the instal or fix database button .
I think this will install the new joomla 2.5 to your server.If you have any issues then please post your issue in the comment.