Like another member said, I think you are getting way ahead of yourself.
College is about figuring out your interests, not only building a career. So take time, a couple of years, to figure out what you are interested in and then study that. A jump from computer science to biology tells me you are unsure about what each of these majors entail, and the respective fields you can get into.
To answer one of your questions, no it is not impossible to get into a graduate school to study biology if you have a bachelor's in computer science. But, without necessary background in biology, chances are you will be rejected.
As far as what university, it depends on several factors: your degree/prospective graduate major, location (do you want to move away or stay local), cost (how much can you afford, or are you prepared to take out loans), your grades. Prestigious schools are, obviously, more competitive to get into and require high grades, high test scores, and unique experiences. But, prestige shouldn't be the criteria to pick a school. Yes, it's nice on the resume, but each school has its strengths and weaknesses, so do some research about each school and compare its prospects to what you want to do.
I went to a state university, but my state university was top 10 research university in the US, which aligned with my interests at the time. It was also local, like I preferred, relatively inexpensive, and had a well-recommended (by working mentors) engineering program. Pick the university that works for you, not the one that is good on paper.