Anybody else?

Hi guys.

I have found this helpful forum and I would like to ask you all a question.
I am a biochemistry student coming to my third year of study. I definitely want to do PhD and hopefully a good one. I am interested in cancer research.
I was the best student biological department in my second year (middle 80's) and third in my first year. I am a mature student. I also got some voluntary lab experience during the summer in after my first year at our Uni, got funding from Biochemical Society for summer project (all of them not connected to cancer research) after my second year and also got a placement year in lab investigating proteins in prostate cancer.
I will also have good reference from my tutors.

The only thing is, my Uni is not a leading UK Uni like Cambridge, Oxford etc. Does this matter at all?
Also I dont think I am very good at selling myself, for example I dont know how to answer a question why do you want to do this PhD. I always wanted to work for cancer research since I was about 20 and it is the most interesting thing in the world but I cant say that, can I? Also my mum and son battled cancer but dont know if to mention this as a reason for my interest. I dont want anybody to feel sorry for me and also this could be taken the wrong way as like I am living in the past.
Have you got any advice?
Do you think I have a good chance?
Thanks guys