We recommend doing a move to a new environment because it has a much easier rollback path if something goes wrong.
Note that these steps will also work for upgrading from TFS 2012 RC to RTM, or RTM to Update 1.
These are the steps to migrate and upgrade to a new environment:
Take the TFS 2010 server offline
Copy the TFS 2010 database backups to the TFS server or the new SQL Server instance. Make sure the URL is accessible from the TFS server via a network share.
Make sure you have access to coffee while it's installing - it could take a while!

After the install has completed, the Team Foundation Server Configuration Center will start
Select Upgrade | Start Wizard

Launch the Database Restore tool by clicking on the link
If necessary, change the Target SQL Server Instance and click Connect
In the Restore Database screen, Browse, then navigate to the folder with your database backups. Make sure all of them are ticked, then click Restore.

Click Close, then click Next in the Upgrade Wizard
Choose the configuration database by specifying the SQL Server Instance and clicking List Available Databases
Check "By checking this box, I confirm that I have a current backup.", then click Next

Leave Network Service as the service account for the Application Tier, then click Next
Check the checkbox to confirm we want to configure Reporting Services, then click Next
Make sure the Reporting Services instance and URLs are correct, then click Next

Update the SQL Server Instance for our Warehouse Database, and click Test to test the connection
Click List Available Databases and check the Tfs_Warehouse database is shown, then click Next

Click Next on the Analysis Services page
Provide details of the TFSService account your reports will run as then click Next

Check the checkbox to say we want to configure SharePoint, then click Next
Choose "Use current SharePoint settings", then click Next

Confirm the details on the Summary page and click Verify
Note: At this point, you may be asked to reboot and start the wizard again. Don't despair - it's quicker the second time!
Once you have all green ticks, click Configure

Have a coffee

Click Next

Click Close, then Close again.
Change the DNS entries for your TFS server to point to the new TFS 2012 server
Note: It's a good idea to get the SysAdmins involved at this stage