My motherboard and CPU are 5 years old. Up until recently I used 5 year old video card (only upgraded to try and get better performance in some other games). So you have to be running some really crappy old stuff to be left behind (my rig runs AH3 just fine WITH the Rift DK2).
To upgrade an old PC (ie replacing mainboard, CPU, RAM) would cost me around $500-$600 max NZDollars. Thats about us$400. And then parts in the US are way cheaper than here, so something like an i3 + 8Gb RAM + GTX 950 is probably in the region of us$340 (according to pc parts builder). If you need a case, PSU, and SSD maybe add another $150-$200.
So an upgrade build $350-$450, a full build $500-$600.