If I have already appeared for GRE Subject Test for Computer Science, do I still need to take the general GRE test?
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.
Which test should I take first(general GRE/GRE Subject Test) and why?
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.