Alright, looking it up, it used DDR-3 1333. You have two options here - newer CPUs will still support DDR-3 1333, so you could re-use your current RAM to save a bit of money, and upgrade it later. There is not a *massive* performance difference between 1333 and 1600.
Here's my suggestion:
Re-use your case, and
temporarily continue to use your current Power Supply. You can repalace it later, and it should hold up to what im about suggest fine.
Re-use your current RAM
Re-use your current HDD if you can (i cant imagine you cant - it should be SATA, just maybe not SATA3. You wont notice a giant difference in speeds. You can always upgrade to a new HDD cheaply later if you run into an issue.
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i3-4360 3.7GHz Dual-Core Processor ($129.99 @ SuperBiiz)
Motherboard: Gigabyte GA-B85M-DS3H Micro ATX LGA1150 Motherboard ($55.98 @ Newegg)
Memory: Corsair 4GB (2 x 2GB) DDR3-1333 Memory ($34.99 @ NCIX US)
Video Card: Asus GeForce GTX 960 2GB Video Card ($189.99 @ SuperBiiz)
Operating System: Microsoft Windows 8.1 (OEM) (64-bit) ($86.98 @ OutletPC)
Total: $497.93
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-05-02 15:32 EDT-0400
Just under your budget, and should run most modern games quite well.
Replace the PSU at your earliest opportunity. Something like this:
PCPartPicker part list /
Price breakdown by merchant
Power Supply: Silverstone Strider Essential 500W 80+ Certified ATX Power Supply ($54.99 @ Directron)
Total: $54.99
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-05-02 15:35 EDT-0400
Will do you fine. Any 80+ 500W+ from a reputable manufacturer will be fine.
Note - there ARE some sacrifices with this rig - you're sticking with DDR3-1333, which IS slightly slower than DDR3-1600; in most cases we're talking single-digit percentage points of performance difference if it is perceptable at all. You're stuck with your old HDD for now, which may be slower (though if it is a 7200 RPM drive, youw ont notice much difference between SATA 3Gbps and SATA 6Gpbs in real-world applications).
But the GPU is strong enough to run almost all modern games at High/Ultra at 1080p, you have a full 8GB of RAM, and the CPU is a fast Core i3 which run rings around your current rig. (both are 2 core/4 thread parts; The COre i5 650 you have CPUBenches at 3144; the Core i3 above at 5473). It should let you stream at 720p or so without too much issue because of Intel's QuickSync technology, which uses the CPU's onboard GPU ("Intel Integrated") to remove streaming overhead from the CPU almost entirely. There are tutorials on how to set it up.