Looking for a binary counter/register IC
Does anyone know of any ICs that are readily available that achieve
  this?


Yes they are called FGPA's. You can make as many registers that you can fit into the FPGA and there is no wiring involved which saves you time. You can get dev boards for less than 30$USD. You could write your own instruction set and design your own processor architecture. You would have to learn an HDL (I reccomend VHDL) and how to simulate, but there are numerous tutorials on the internet to get you started. And you'd have a useful skill when you were done.


  Alternately, if anyone has ever done this before, how do you make the
  PC? Is it a single chip or do I need to create a separate PC circuit?


You probably won't be able to put together a CPU, that is too detailed. It has been done, there was a guy that did a project similar to this on a transistor level took him a few years. Even with integrated logic and a simple CPU this would take a long time.

I built an 8088 computer by wirewrapping, which included connecting the memory, ROM and I/O. This was for a university course, it takes about 6 weeks and takes about 40 hours a week. This sort of thing takes time, even without mistakes you will spend a few man-weeks implementing it. 

The 'way' you do this is draw up your processor architecture with registers and wires, then find discreet logic IC's that will perform the function of each one.