If he has an AM3+ MOBO, it SHOULD support an updated processor and memory. But you should check the MOBO specs to be sure. If so, then you can rule out getting another board which would save you some cash.
Also SSDs are pricey, sooo… Your current HD still looks like it’s decent. Might be able to save some cash there as well by keeping it. That would narrow it down to a new Processor, Memory and Video card if you want to be more conservative and stay within your price range.
The processor you listed runs around $110, add DDR3 memory around $40-50 dollars for 8GB, then throw in a HD6850 for about $150 and you are within your target range, Might even be able to go a little higher on some of the parts if your willing to go up to $400.
I would also take a look at your current PSU as upgrading some of those components will change your power requirements and you will want to make sure your current PSU can handle it. If not, then you need to add the cost of a new PSU to the list.