Minecraft will not run on a Chromebook under default settings. Because of this, Minecraft's system requirements list that it is compatible only with Windows, Mac and Linux operating systems.
Chromebooks use Google's Chrome OS, which is essentially a web browser. These computers are not optimized for gaming. In order to play Minecraft on a Chromebook, you would have to "root" your device to access the Linux kernel (though that defeats the purpose of a Chromebook). If you would like to do this, a number of guides are available on the internet, but please be aware that this does not ensure full compatibility with Minecraft.