The easiest way is to have an internet ready smart TV that has an Amazon Prime APP. Then you just run the app on the TV and stream away. Other than that, you must get/have another device that connects to the internet and has an Amazon Prime App or fully functioning browser. You then use that device's video/audio outputs to connect to your TV. Similar to how you would connect a blu-ray player to your TV. There are many devices that can do this: Gaming Consoles, Mobile Devices, Streaming Media Players, AV Receivers, Blu-ray Players, Computer/Laptops, etc. You just need to make sure the device is compatible with Amazon Prime. Which is basically two way, it can run an Amazon Prime APP, or has a fully functioning Web Browser.
The complete list of compatible devices is at:
http://www.amazon.com/gp/video/ontv/devices/ref=hp_3757_compatible