aymanhasan93 already has ddr3.
The board for the i5 2320 isn't going to be cheap at all.
As you're on a budget you can't expect top-notch gaming but the fx6300/50 is a very good processor. You can play today's games at 1080p and stream with ease and do so for a long time to come. In future games will use directx12 so that's even better for this processor unlike an i5 as 6 cores is the optimum.
You can get the fx6300 cheaply on ebay. Ddr3 1333mhz is also cheap there too (the fx6300 uses up to 1866).
I've read that people with i3 and i5 processors have experienced stuttering in certain games. This means all cores/threads are at 100% too often, which isn't good for the processor's life.
Intel processors will suffer from security flaws like Spectre
https://www.youtube.com/watch?v=NArwG6yaWJ8&feature=youtu.be
AMD's new processors could too so I'm sticking with their older processors like the FX6300.
This video shows that the fx6300 can play Assassin's Creed Syndicate at 1080p very well and no core gets near 100%
https://www.youtube.com/watch?v=vfr0ktiGTAI
(I've seen 1080p YouTube videos where even on the older slower AMD 4-cores with this game none of the cores reach 100% and that's with screen-recording software being used. In one of the videos the motherboard used DDR2 RAM which slows the frame rate down more than on DDR3 and yet play is still smooth. This game is a good test because the play takes place in old London where there are a huge number of buildings in detail plus many people and trees with all their leaves and shadows cast by all those objects, plus small objects like puddles on the ground with reflections and smoke from chimneys and horses moving their bodies and heads and birds flying around and there's also debris and litter flying in the air. Also the clothing people wear can flap around as people walk and there's boats on the river with some carrying cargo like logs of wood and lamp lighting and burning fires with puddles on the decks and reflections in that water and smoke coming from the boat funnels and plumes of water behind the boats as they move through. The main character climbs to the top of tall buildings and you can see the whole city. That's a PHENOMENAL number of objects and this game isn't even DirectX 12.)
https://www.youtube.com/watch?v=eNvaJJMEFco shows an older AMD 4-core 3.ghz running the game well at 1440p & 4k.
If you need a quiet good processor cooler try the Cooler Master TX3 or the better $30 Be Quiet Pure Rock Slim.
One feature of the FX processors is for the processor to overclock itself if any core reaches 100% load. In the BIOS look for “Advanced Clock Calibration” which should be set to Auto (on an Asus motherboard this is in the CPU Configuraion menu). The item below it is called “Unleashing mode” and this should be enabled.
Benchmarks are flawed because they use Intel's compiler. They're bribed by Intel to make AMD look far slower.