Hi Andrew,
I would say the best thing you could buy is a core i-7. These are known for better multitasking purposes since they handle threads a lot better as opposed to the i5 or i3. If you can get something with a big cache on the chip (8MB cache > 6MB cache), will help you a lot since the computer architecture will be able to keep more program and data memory within the processor making it faster and more efficient. Of course as always get ram (8gb will work fine, 16gb will work better.
However when it comes down to it, the operating system is perhaps the most important part. Since you need to adhere to these real time constraints, stripped down versions of OS's tend to work better since there's less overhead in the background processes running. I am assuming you are using linux? One of the minimal distro's should work plenty fine.
Just know that people have successfully run fully functioning CNC machines on Single board computers with linux CNC on there just fine. Machinekit is a group of people that helps build the linux kernel for real time constraints.
I am fairly new to CNC also and have been a lot of research for it (electrical engineering student) so i would probably get someone well versed to confirm what I just said, or suggest something else if I am not in the ballpark of what should work.
Good luck with your machine!