Chromebooks are a bit too limiting in my opinion especially since it is not compatible with Windows software. I would get the following 15.6" laptop with a 7th generation i3-7100u, integrated Intel HD 620 graphics core and 1080p screen for $350.
https/www.amazon.com/Acer-E5-575-33BM-15-6-Inch-Notebook-Generation/dp/B01K1IO3QW/ref=sr_1_3?ie=UTF8&qid=1520621743&sr=8-3&keywords=i3-7100u
The Intel HD 620 is good enough to play games that are not very graphics intensive especially if you drop the resolution to 1366x768. Though I would recommend you install stick of 4GB DDR4 RAM if you wish to do so to bring total RAM to 8GB... and two sticks of RAM can improve game performance (when using integrated graphics) by 10% to 20% depending on the game.
For example, Skyrim would be playable at 1366x768 resolution using default settings. I am sure if you tweak the graphic settings and the SkyrimPref.ini file you can play Skyrim at 1080p resolution. That's base a video review of Skyrim being played at 1366xx768 resolution on a laptop with the i3-7100u. As well as my own experience tweaking Skyrim to run at 1600x900 for a friend's laptop that has a 4th generation Core i5 CPU with the Intel HD 4400 from 2013.