The new Amazon Kindle Fire HD tablet runs a heavily customised version of Android Jelly Bean with restricted access to the Google Play App store.
To install the VPN please go to "Wireless" menu.To get to the "Wireless" menu you have to swipe down from the top of the screen select "Wireless" . From here it is the same as any other android device.
Wireless > VPN > Add VPN Profile > Type L2TP/IPsec PSK VPN
Once installed you will need an Amazon account to gain limited access to the Google Play App Store. The billing address and payment method needs to be set to your target country ie UK, US, Australia or Canada to be able to download country restricted apps such as BBC iPlayer.
What does restricted access mean? The lack of Flash plug in meant no access to YouTube and utilities such as ITV Player and 4OD due to it being unsupported by the Silk internet program that is installed as standard. For example for the UK BBC iPlayer, Netflix and Tune in Radio are available. There are workarounds out there to gain access to all Google apps for the more adventurous.