I went slightly over your budget with this build but its a value gaming build so it should work well for you.
https://no.pcpartpicker.com/list/LDgMzNIgnore the compatibility warning, this motherboard and CPU do work together, this build is almost identical to this build by linus tech tips, you could follow it to build your computer, or even change parts to match his even more closely.
https://www.youtube.com/watch?v=P8BEDv_p8TQ