Qualifications
Do you like school? Well, if you want this career, you'd better at least be able to fake it well.
Getting a bachelor's degree is a no-brainer—or might as well be, because it's not going to get you all the way to this position. Even a master's will only get you on the second-string roster. To really be the top of your field, you've got to get that PhD in cancer biology.
The doctoral degree is the final destination for the serious cancer biologist for several reasons (source). First, depending on the track you take, it may be the only official piece of paper that actually says the words cancer biology. Second, and this is the important part, getting a PhD means that you have dedicated every moment of every day to learning about, studying, and figuring out how to eradicate cancer.
The extra five to seven years it'll take to get that doctorate is proof of your commitment and your expertise. People will want to use that knowledge, and will give you access to all of the equipment and staff you could possibly need.
And then there's the money. Whether we're talking your funding or your salary―which are probably related―a PhD in cancer biology will give you access to lots of money. Makes all that extra time spent in the library on Saturday night worth it, right?
If you're considering this as an option, then you've got to have a firm grasp on math and natural science. Obviously biology is in the title, but everything from anatomy to chemistry to physics to trigonometry will help you in the long run. Anything that helps your problem-solving and critical thinking skills is good, because cancer is one big unsolved problem.
Besides your brain skills, your people skills will be put to the test. As part of a team, you've got to be able to communicate; otherwise you're just a bunch of silent super-geeks standing around a lab.