I received similar grades (4 A*, 4 A*, 1B and 1C) and my school have been very supportive of me seeking to apply for Oxford! There were a few people in my year who got 11 A*+ (three, I believe, of whom only one wants to apply for Oxbridge - Cambridge for German & French). I also got rejected from the Oxford Pathways today, so we have that in common too!
Our Oxbridge programme has a benchmark of 5 A*, but the teacher made an exception for me. I'm sorry your school haven't been supportive. What I would recommend is that you look into any of the academic competitions - I'm personally looking to enter two history essay competitions - as well as getting involved in extra-curricular activities that are relevant to your subject (I'm volunteering at my local National Trust property, for example). Perhaps if you talk to the teachers some more they could be a bit more accommodating.
Depending on your subject, you can be almost guaranteed to get an interview at Cambridge if you get your A's, get predicted the right grades, and have a decent personal statement, in my experience. It does really depend on your subject though, which would it be if you don't mind me asking?
(I've also heard Cambridge care a lot less about GCSE grades than Oxford, for what it's worth, I don't know if that's just hearsay though).