An Amazon Fire Stick has to be plugged into a TV's HDMI port, so without a TV you won't be able to use one.
Once you've got one plugged in though you can use the apps on it to access free services (such as BBC iPlayer and Youtube) and, if you pay the subscriptions, services such as Netflix and Amazon Prime Video. (Netflix costs £5.99 per month for Standard Definition, £7.49 per month for HD or £8.99 for Ultra HD. Amazon's basic service costs £5.99 per month, with their 'Prime' service costing £7.99).
You don't need a TV (or an Amazon Fire Stick) to access all of that stuff. You can watch it all on your computer anyway (although, of course, you'll still have to pay for the subscription services). All that an Amazon Fire Stick does is to enable you to watch on your TV instead of your PC or laptop.