Yes, sort of. You have to have an Amazon Prime account to get the free unlimited video streaming, however even with a Prime membership, for some recently released movies and new TV shows single episodes or whole seasons, there is an additional charge. But content eligible for free unlimited streaming is clearly marked with the Prime logo.
If you do not have a Prime account, you can still buy or rent video content but dont get any free content.
The subscription costs $99 per year but in addition to movie and music streaming (similar to Spotify or Pandora) and some other perks like 1 free Kindle book to borrow per month, you get free 2-day shipping on Prime eligible items and for frequent Amazon purchasers like me, thats the real advantage.
This gives a pretty good overview:
http://www.tomsguide.com/us/what-is-amazon-prime,news-18041.html
I order enough from Amazon, all year round but especially at Christmas (I do all my Christmas shopping on Amazon) that the free shipping alone more than makes up for the cost of the subscription.
I dont have a Kindle but am considering getting one. I have streamed some movies to my laptop but would need an internet capable TV or Blue Ray or game console or a device like Amazon Fire TV in order to stream to the TV.
If your laptop has video out and the appropriate video in on the TV, you can just connect the two. HDMI works well for this.
I have a MacBook Pro 15" about 5 years old. I have a 5 meter cord that takes the external video output from the Mac and connects to my 32" TV via HDMI. I use a separate audio feed since it does not provide that on the Mac output. I was just watching "BlackList" on Netflix here in my Bedroom office.
Newer Macs (and I suppose other Windoze types) provide direct HDMI, YMMV!
DVI to HDMI