Playing unblocked games on a school Chromebook isn't something I'd recommend. School computers usually have restrictions for a reason, often to ensure students stay focused on educational content but also admins have likely implemented monitoring tools, so they may be able to tell if you ran something that wasn't approved.
That said, you may be safe to enjoy browser-based games on your school Chromebook by launching the Chrome browser and running games that can be seen as educational in a way.
There are many gaming platforms and websites that playable games directly in your browser, with no need for any downloads or installations. Opt for HTML5 or WebGL games, since they are designed to perform well on Chromebooks.
Two good ones to get started are crazygames.com and kongregate.com.
Now, if you want to take it up a notch, try a true learning game that teaches you how to code, try these:
CodinGame is a challenge-based training platform for programmers where you can play with the hottest programming topics. Solve games, code AI bots, learn from your peers, have fun.
You can play unblocked games on 2playerunblockedgames.com. They have a collection of games that are not restricted in schools. PS: the site is safe to browse.