How can I increase my chances of getting cancer?

Honestly, it's all down to luck. The factors that define what you get and where are completely random. First off, when you start, the computer "rolls a dice", and depending on what you get between 1 and 7 defines your luck (item find, possibility of getting curses etc.). 1 being "Curse of the dark" every time, 7 being amazing items in every gold room, arcades on every floor, etc.

Also there are items that certain characters start with like the 6-sided die, that allow you to re-roll gold room and shop items.

However, the best place to get Cancer with the highest drop chance is Mystic Slots. Other than that, cross your fingers and pray to Pinocchio.

Cancer is a trinket slot item like The Tick, or Cain's Eye, or The Polaroid.

The effect it has is to reduce the delay between shots - this is independent of the "Tears" statistic that governs fire speed, so as you get faster tear shots this static reduction in delay is a greater % of the remaining time and your DPS increases geometrically.

Trinket slot items are gotten from room drops, chests and the Fortune Telling machine.

The best way is then to play as Cain - the Luck Foot increases your chance of getting items from room drops and chests, but most importantly it removes the "Fortune" part of the fortune telling machine (the messages across the top that say things like "Steven Lives!"). This means that every time you put a coin into the machine you'll be getting a) a tarot card b) a soul heart or c) a trinket. Given that the chances of winning the fortune slots is fairly low for other characters this is a huge huge bonus.

Fortune Telling machines spawn most often as parts of Arcades. Arcades only spawn on even numbered floors/XL floors, and only if you have 5 or more coins.