Can I pursue a PhD in Computer Science after completing BCA and MSc in CS from India?
Yes, you can definitely.
Yes, MSc in Computer Science qualifies you for most PhD programs. However, the specific requirements might vary depending on the university.
But do I need to take any entrance exams like the GRE or something?
It generally depends upon which University you are applying to. For universities in the U.S. or Canada, the GRE is often required, especially for computer science PhDs but english proficiency tests like IELTS or TOEFL are a must where the teaching medium is english.