Oh boy - this is a hard one due to COH2 being both GPU AND CPU intensive.
CPU wise you're constrained to a minimum of an i5 or FX-8350 (If you're AMD inclined) - and Intel prices really haven't dropped, i5-2500k's still hold their value, so you really can't get a cheap option here. Expect to pay about $300 +/- for a good CPU/MB combo.
RAM - I found 16GB to be a "sweet spot" for COH2 - 8GB was noticeably slower.
GPU - Now here you can find value. I'd recommend picking up a used 7950 (which is re-badged as a 270X) for +/- $100 on ebay, or a 7970 (re-badged as a 280X). No reason to buy the "New" card when the old one is just as good, if not better in some ways given the overclocking ability of the 7950 and 7970.
An SSD, while nice, isn't a necessary component for a "starting" computer - but for $99 you can get a Crucial Mx-100 256GB which would be more than enough for an OS and several game/programs.
Still - its really hard to get a "cheap" performance COH2 system.
So assuming $300 for the CPU/MB, $100 for the GPU, $100 RAM, $80 PSU (700W + Seasonic etc), $80 Case (depending on taste) and $100 SSD - you're looking at $760 for a very strong system.
It's true you'll need a little more than the limit for the system, but please tell us how coh2, a 32-bit program, can address more than 4GB? If your windows is choked by other programs running along with coh2 that is something else. 4GB is all you need to MAX coh2. I haven't checked in game memory usage, but I have a feeling it's much less than 4GB. Just at the title screen it only uses 700MB on my system.