Lets get some things cleared up first, there are NO keys Blizzard has stated that they aren't doing keys they will flag your account. That being said lets move on.
- If you have no install/play button chances are.. You're not in, but don't worry we aren't done with the tutorial!
- If you have the install button, or if you have the play button, there is a chance you could be in. But this is not a guarantee! There is a bug/exploit allowing people to download the client but this does not mean that they are into the beta. The best possible way to know you're in the beta is to check your battle.net account status.
- First you need to log in. (Just type battle.net and log in) Once you do that you'll be taken to your account summary, here you will see all of your game accounts. If you by chance are in the beta you'll see it here. There will be a new game called Overwatch and underneath it you'll see the word beta. This means you are in!! It will look a little something like this.
TL;DR check your account status through battle.net
I have also made a video explaining all of this too that you can check out here
I hope this help you all, and I wish you all the best of luck getting into the beta! And for those of you who do get in... Congrats! I hope Overwatch is a blast!
Here is a cool picture from pony-76 https://i.imgur.com/lh1mkAa.png This is what it should look like if you get in :)