No, your CPU will not need upgrading, especially as you're working on a budget. I've used an E2180 (inferior to your E6400) & a HD4870 graphics (GPU) with very good results.
Also, please be aware that the E7600 is one of the latter-day Core2Duos & the PB motherboard will most likely need a BIOS update in order to support it. It's very unlikely PB are still releasing newer BIOS versions 2-3 years down the line.
Yes, the GPU & PSU would need upgrading as you know. However, first check you have the necessary physical space in the case to fit them.
Today's GPUs (including the HD4850) are quite large length & breadthwise. These are the case measurements you want to check for the GPU click here - let us know what these are.
Measure the dimensions of your PSU & let us know. If it's a standard-sized one, then any other standard one will do.
So for a start, let's see if we can determine if the new components will fit, or will you also be needing a larger case. Then we'll discuss the actual upgrades, how much they'll cost & decide whether a PS3 / Xbox may be the better option. G