Hi Tom.
The Core i5, i7 and AMD phenom series all have different models and more importantly generations (although the exception is the x6 which is only in the phenom II series next gen of processors I suppose). The amount you are looking to spend may be more than what you need based on the information you gave. If you could be more specific about the applications you are going to use and hve running simultaneously, I could provide a more informative suggestion. However, just based solely on this information (and the assumptions I am going to list below) I can make a few recommendations for you.
Assumptions:
No 3d rendering, animation, modelling etc (no use of programs like AUTOCAD, MAYA, 3dsMax)
Gaming limited to games made until 2006 utilizing upto directx 9.0c
Purchaser (i.e. you) is in/can purchase from the US (local prices in another country may vary)
Some definitions:
Hyperthreading: an Intel-proprietary technology that aims to improve parallelization of computations and works by the operating system addressing two virtual processor for every physical processor present and shares the workload between them when possible.
A little background information:
When going pre-built, the model of the processor is important; a core i5-661 is not the same as a core i5-2500, and so on. So their features may be different in addition to their processor speeds.
For example, the core i5- 6xx series processors have onboard intel hd graphics, which, although not good for top of the line games, are more than adequate for less graphic intensive games... (certain models within series may still have certain features disabled, such as the 661 not having hyperthreading).
The core i5-6xx series have 2 cores and some models have hyperthreading, while the core i5-7xx series have 4 cores and no hyper threading. i5-2xxx series have 4 cores and no hyperthreading but are faster in terms of CPU frequency and feature builtin graphics.
CPU Suggestions:
Based on the information you have provided and the cpu's you are undecided between, I would either recommend a core i5-2xxx or a Phenom II X4 or X6 cpu (although the x6 would be underutilized for your currently mentioned usage). The Core-i5 2xxx series is the next gen processor based on sandy bridge architecture and features 6 MB L3 cache (good if you want responsiveness when starting a program you frequently use) and also has builtin graphics.
As I said, the series of the processor is important so you should keep that in mind. If you want, look for some systems online and tell us which are the ones you are considering and we can provide a more guided suggestion.
Here are some systems that I found to meet what you want and you may like (these systems all feature good graphics cards):
http://www.newegg.com/Product/Product.aspx?Item=N82E16883227321
http://www.newegg.com/Product/Product.aspx?Item=N82E16883229247
http://www.newegg.com/Product/Product.aspx?Item=N82E16883229248
Please note that I have never bought anything from these brands so I have no idea as to the quality of their products or customer service. The problem with most pre-built systems is that other than the processor, you really have no idea as to what you are getting (well, ok maybe the case too but thats only because it is on the outside and visible). I would recommend you getting a custom built PC if you can (maybe ask a friend who can to build one for you?).
If you do decide to go down this road, there is an excellent article on this site that has a system just for your budget:
http://www.newegg.com/Product/Product.aspx?Item=N82E16883229248
If not, then when looking for a PC model, try to do some online searches on forums to see what users are saying about that model of PC. They can be fairly informative.
Oh, and one more question for you: why are you not considering a core i3 system? the core i3 is the mainstream processor and more than capable for what you are planning to use it for.
I hope I was able to answer your questions. When you decide on the CPU model, if you want help on deciding a system (in terms of upgradability, suffieciency of RAM, graphics card, etc) I would be more than happy to help.