There is 2 ways of doing that. One is by uploading a theme on WordPress panel, and the other one is by uploading a theme through FTP.
Before updating anything, we recommended do a backup of the site.
In our blog has a tutorial about how to do a backup via cPanel, and another one about how to do a backup via phpMyAdmin (portugues only).
Updating a theme through WordPress panel
First you need the file zip of the new version of the theme you want to update.
E.g.: the Simples theme has a file simples-v1.1.0.zip
After you have a theme file in zip format, log in to your WordPress panel and go to Appearance and then click on Add New, and on the next page click on Upload Theme.
On the next page, click on the button to upload the theme file.
After that, WordPress will show a message saying that you already have a theme with that name, and will ask if you want to replace it.
So you just need to click on the button Replace current with uploaded.
Updating a theme through FTP
After you have a theme file in zip format, unzipped it first. It doesn’t work if it is a zip file, you need to unzipped it.
E.g.: the Simples theme has a file simples-v1.1.0.zip that when you unzipped, it just a directory called simples.
You will need to have a FTP program installed on your computer.
After that, log in with your site FTP access on the program.
Then go to directory wp-content/themes/ and upload the theme you just unzipped, replacing the older one.
If you don’t know how to use a FTP program, we recommend you install the theme through WordPress panel.
On the blog we have a tutorial on how to use FileZilla (portugues only).