My computer is very similar to your dad's computer. But mine is running with a 1.2GHz Intel Celeron processor.
It would be difficult to upgrade such an old system. With the exception of the hard drive and the case, the motherboard and almost everything else attached to it are completely obsolete and unable to handle new components.
Also, the recommended system requirements for your new games are awfully high.
In terms of maximum affordability, try something along these lines...
And finally, building your own computer is much more cheaper than buying a completely new one. Plus, you could re-use some of the old parts from your existing computer.