Will I be able to get into Oxford or Cambridge University to study computer science if I have A* in maths, computing, biology and chemistry?

Yes most definitely! For Oxford you need something along the lines A*AA with A* in Mathematics, Further Mathematics, Physics or Computing. For Cambridge you need A*A*A with similar subjects to Oxford (Mathematics is essential)

The short answer: you need these grades but they are not enough. Having certain number of A* is one of the minimal formal requirements. After applying for CS (at least in Oxford) you have to write a test. Many candidates are rejected at this stage. If your marks are decent you will be invited for an interview. At this stage there are about 3 candidates for one place. So you have to perform well on the interview. BTW all offers are conditional: if you are given an offer but failed to get your expected A*, then your offer might be withdrawn. P.S. Extracurricular activities might help you to get the interview invitation if you are very close to the cut-off, but they don't play much role after that. They are very important at the American Universities.

Based on my own personal research going through their websites, for the A level score qualification that is more than enough already, there are also other known factors taken into account such as your academic and non academic achievement(s), participation in extracurricular activity and social work. Assuming you also made a proper essay, you'll get an interview or even short-listed, now then it's based on your luck and communication skill.