Login to your server and backup the 'default1' directory found in the 'accounts' directory. This step is not essential, since the update does not delete any files, but it seems some customers (or kind of weird FTP client) tend to delete the 'default1' directory even if they are not supposed to do so. Just in case you can also backup your database.
Check if you can unzip zip files directly on your server. If so, upload the package zip file to your server and unzip it so it overwrites the old ones. If you can not unzip it there, you have to unzip it in your computer first and then upload all files to your server (using Binary transfer type) so it overwrites the old ones.
When you are updating your Live Agent, your customers (and agents too) will see a message like this:
Now open your main Live Agent site and click the link Installation and Upgrades or navigate to https://yoursite.com/liveagent_directory/install/
Select the language in which the update steps should be displayed. This language doesn't affect anything else then the text of update steps.
After choosing a language you will be presented with a check requirements screen which checks your server configuration and whether it meets the requirements.
Next screen will remind you of the importance of backups and will tell your previous version and the version to which you are going to update. When you are ready you can click "Start Update"
You will see the list of update steps slowly filling in (it can take some time depending on the age of previous version). When all is done you can click Next to finish the update.
When the update is finished you will be presented with a screen like this
Note: In case you encounter any kind of problem or issue after update, it is recommended to first clear your browser's cache. Files from the previous version might be cached, and might cause errors due to incompatibility with the current version.