Usually, we can upgrade win 8 to win 8.1 via windows store.
But for this kind of major update, to perform a clean install is a better idea.
We can follow below steps to install win 8.1 with win 8 key:

1. Download Windows 8.1 Image from MS website
2. Create a USB installation disk
3. Install windows 8.1 with windows client key
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4. After the installation is finished. Press Win+Break key, and change windows 8.1 key to your windows 8.0 key.

System will be activated automatically.