First off, you will need to backup your database & theme files. Since we're replacing and updating theme files, there's a chance something could go wrong so it's best to play it safe and have a backup.
Use whatever method you feel most comfortable with. You may want to simply copy your entire WordPress directory to your computer using FTP and backup your database with a plugin. You may have a VaultPress account or a membership with some other third-party backup site that does all the hard work for you.
Now you can start upgrading. Go through FTP client to your server, find directory /wp-content/themes/[theme-name] and remove it or rename(safer method, we will remove it soon).
Then, go to Appearance > Themes > Add New and upload archive with new version of your theme. Activate it and check for errors.
If everything is fine – you can completely remove folder with old theme version from your site.