I did a Thesis upgrade recently for a new customer, because some core files in their Thesis installation had been altered.
Altering the core files of your theme, or of WordPress, will always generate problems at upgrade time because your alterations will be over-written.
That means the functionality that you created through the customisations will be lost and you'll need to re-create it.
So where you customise a theme that's not a Premium theme you should create a child theme or, if it is a Premium theme, use the settings panels or custom stylesheets.
That way your customisations will be preserved through upgrades.
Although it's very straightforward, upgrading Thesis is not a one-click operation.
The DIYThemes site has step-by-step instructions on how to upgrade Thesis - they're in the 'downloads' page but you'll need to login to see them.
But, since I have a number of Thesis sites I'm managing, I've added a few extra steps that help me to be more efficient:
Step 1 - Preparation:
Download the custom folder from your live (current) installation of Thesis to your machine.
Depending on the customisations you've done, you should have at least 6 items: layout.css, custom.css, custom-functions.php, images folder, cache folder and rotator folder.
I'm running several Thesis sites, so each has its own folder on my local machine so I can keep the settings for each site separate, and I download the relevant custom folder to the relevant local folder.
Download the 'All Options' file to your machine from the Manage Options screen. This is not strictly necessary because your settings are preserved through upgrades, but I do it as a backup.
If you're running several Thesis sites download the relevant All Options file to the relevant folder on your machine
Step 2 - Download the new version of Thesis:
Download the zip file from the DIYThemes site.
If you're running multiple Thesis sites, extract a copy of the new version of Thesis to each local folder on your machine.
Step 3 - incorporating your customisations
(Do this in each local folder if you're running multiple Thesis sites)
Copy the current (live) version's custom folder (that you downloaded from your site earlier) to the new Thesis folder.
Remove the custom-sample folder, so that you're left with the custom and lib folders in the new version.
If you're updating from a version of Thesis prior to 1.6, copy the contents only of the downloaded custom folder to the custom-sample folder and re-name it to 'custom'.
Upload the new version of Thesis (which now includes your downloaded custom folder) to the wp-content/themes folder on your server
When the upload has completed click on the new version (from your FTP client) and then click on the custom folder to reveal its contents
You should see 6 items - 3 files and 3 folders. Right-click on the layout.css file and set its permission to 666. Next, right-click on the cache folder and set its permission to 775.
You're now done for your FTP connection and you can disconnect it
Step 4 - Activating the new version:
In your WordPress Admin area, go to your Appearance>Themes screen and click on the new version of Thesis. Click 'Activate' in the top right when the preview is displaid.
NOTE: the preview version may look different from the way your site normally does. This is because the options haven't been saved yet, so ignore the differences.
After clicking 'Activate' you'll be taken to the Thesis Site Options screen. Click the Big Ass Save button, then go to the Design Options screen and do the same.
You should not need to import the All Options file you downloaded earlier, but check your site to make sure everything is as it should be. If there's a problem, importing your All Options file is the best place to start.
In this case, go to the Manage Options page and import the 'All Options' file you downloaded earlier to your machine. Click OK to confirm the import.
After importing your All Options file, go back to the Site Options and Design Options screens and click the Big Ass Save button.
OK, that's it - you're all done.