A lot of universities in the US and the UK offer PHDs in computer science. But the ability to enroll to some of the elite CS departments requires you to convince the university that you are a possible candidate for their PhD program. The best is to evaluate your current academic record, what you have done in the field of computer science. Involving in a PhD program is a extreme investment. So first think about the future expectations of you and start on applying for the universities that you think you’d fit in. It is very helpful if you can have a chat with a academic figure of your local university.