If I have already passed a GRE Subject Test for Computer Science, do I still need to pass the general GRE test?
Which test should I take first (general GRE or GRE Subject Test) and why?
Short answer: Likely, yes. If your school asks for the general GRE, then you need to take it. Without the general GRE scores, your application would not be considered complete, and, thus, may never get forwarded to the department's admissions committee. If in doubt, call or email the department(s) you are applying to and ask.
It does not matter. Usually the general GRE is offered more frequently than the subject GREs, so many people take it first. There is no reason you have to do it this way. Take them in an order that is convenient for you, but do not wait until just before your application deadlines. Your scores will not get to your schools in time.
The tests are essentially independent, so you can take either one first.
Some schools want the subject test, but others don't care. This varies from one school to another. I got a PhD in computer science from a good school, but I never took the computer science GRE subject test. On the other hand, I did study a bit to improve my scores on the GRE general. If you want to know which tests to take (and how heavily they'll be weighted in your application), I think your best option is to contact the school directly (or just read the application checklist on its website).