GTrusted

If you want to use the Amazon Fire TV (2015) to play 4K to your LG 4K Ultra HD Smart TV with webOS 2.0 (2015), it's pretty simple.  All the HDMI ports on the LG 4K TV support 4K so just plug your a HDMI from your Fire TV to one of these HDMI ports, and you are good to go.

Oh yes, remember to find content that is actually playing in 4K like in the case of NetFlix which can show you that the video being played is 2160p.  Most of NetFlix  4K shows are labelled as Ultra HD 4K.

Also, make sure also you have enough Internet bandwidth (ideally over 20Mbs) to support 4K streaming.