After purchasing a Rhino Upgrade key you have two options to access the latest version of software.

Before anything ensure you install the latest version of Rhino here.

The first, and simplest, method is to open your newly installed software and when asked for a key add your new one. After that it should recognise it's an upgrade key and ask for your old key.

We often see this method fail so follow the below for a backup method. 

First head to your licence page here - https://www.rhino3d.com/licenses 

If the licence you are upgrading is in a team go into that team otherwise head into your 'Personal Licences'

From here you have an 'Add Licence' button. Again enter the new key and that should add it or ask for the old key. 

Then head back to the Rhino application and again log in and it will then recognise the key as active and valid. 

If the issue continues, get in contact with you reseller or McNeel directly.