I used to play Minecraft on Chromebook (both versions!), and I've seen some incorrect info in some of these comments, so I want to clear this up.
The official way (and way I recommend; more on that later) of playing Minecraft on a Chromebook is by buying Bedrock Edition from the Google Play Store. This purchase is separate from buying it from the Minecraft launcher, Minecraft website, or PC. Buying it on the
Google Play Store links the purchase to your Google account, so you can play it on Android as well. If you already purchased a copy of Minecraft on a Microsoft account, that Bedrock you have already bought won't be playable on Chromebook. If you bought the game through the Google Play store already, make sure you are signed in with the same Google account that you purchased it on.
Some Chromebooks are able to run Java Edition as well, using the
Linux terminal. Doing this will allow you to play Java Edition through the launcher, so if you purchased the game through a Microsoft account, this will let you play it. To do this, download the Linux launcher on
their website, then move the file to the Linux folder, and open it to install it. You cannot play Bedrock this way on a Chromebook. Unless you have a really high end Chromebook, this likely won't work well; it'll be really laggy and buggy. I don't recommend doing this, because it isn't as straightforward to setup and doesn't perform well because of Chromebook's limited RAM.
If you have any follow-up questions, I can answer them. Hope this helps!