You can make many many combinations of gaming pc's with Intel processors for that kind of money... the biggest question is... what games will you be running and what type of graphics quality and frames per second are you looking for?
A Intel Core 2 Duo 2.4Ghz system with 3 or 4GB RAM, power supply upgrade, and decent video card could work out well for you if the games you will be playing are ok with a Dual-Core processor.
You could even get an Intel Core Quad system with 3 or 4GB RAM, power supply if needed, plus video card for less than $450 in some situations for games requiring a quadcore cpu.
If you want to go with modern vs a refurb computer with upgrades, for $450 you will find that the cost in your CPU will directly offset the GPU processing power that you have money for.
So you could get a core i3 or i5 system built up with 4GB RAM, + Windows 7 or Win 8 + whatever video card fits into what is left for money and a power supply if one is needed or not.
*** Do you have an existing system in which you could simply reuse HDD, Optical Drives, Case, and Power Supply from in this new build? Cost cutting by reuse can allow for you to get more with your money, however when reusing a power supply be sure that it is plenty for the system to run.
I have repurposed hardware for years mixing and matching to cut corners in money that needs to be spent, for example I have a hard drive in my one gaming system that is a 2009 build and the secondary hard drive in it is a 164 GB SATA 1.5 drive paired up with a SSD, so its fast for everything that needs to be fast and this hard drive with a date on it of August 2004 is still chugging away and has plenty of speed and storage for the needs of this system + 50,000 operating hours and counting. *Note: The only problem you could run into with reuse of hardware is if for example the old system is too old and so the HDD is IDE and the new system is a SATA motherboard without an IDE port. But generally unless your really scarce of cash, its best to install a SATA drive for greater storage capacity and speed than trying to make a modern gaming system with a 160GB IDE HDD, which I did for my wifes system, but it was only because her system had a IDE ATA100 port on the motherboard and 160GB is plenty of capacity for her and its a secondary drive in which her primary drive is a SSD for making her Core 2 Duo 2.4Ghz E6600 system run fast with quick boot and load times for everything that is on the SSD, while personal data, music, pictures, and office applications, and some games are on the IDE drive which is an ok pairing for what it can handle.
