although it Isnt a huge budget, You CAN certainly make Something "doable" happen with that budget. My suggestion would be to Avoid "performance" parts i.e. SSD's, fancy PSU's, coolers, cases etc. I have done what im about to suggest myself, and although it ISNT optimal it Can get oYou up And gaming for Less than $400.
so if I was in your shoes, I would first Buy one of these...
HERE
it comes with an operating system,an i5 2500 NON "K" (a very capable Gaming or otherwise CPU) 8Gb's of RAM,a 500Gb HDD ,and even a cheap mouse and keyboard, ALL for
$170
that leaves
$230 ,
I would then buy a
750ti ( less than 2 inches tall and under 7 inches long)the ones i would suggest require NO external power, and can run off of the Power provided by MOST pcie lanes (45-60 watts) I have ordered a Low profile one to fit into my nephews optiplex 7020 SFF, and the case in the link i provided is NO smaller than that.
that build with a 750ti, and an i5 2500 will run MANY games in 1080p perfectly fine, and beyond. I know for a fact it would even run GTA5, since i have run gta5 on an i3 4160 with a 1Gb 6950, and the build i recommended is a stronger gamer than it is.
Your then finished,buy, and install some games with the remaining $110.00 and have fun
**NOTE** in SOME situations, if heat becomes too much in summer months, removing the side panel works wonders, but what do You expect for a Sub $300 Gaming PC
oh, and I just saw the whole title, I realize this isn't a "quality" gaming PC , just a capable one. sorry if this doesn't match what you wanted.