So I started the project today, hoping to get the camera showing some images of text on the computer. I succeeded ... partly. The camera seems to struggle with differentiating between anything but extremely bright and dark objects. In the end, I was only able to recognise bright sources of light (such as the ceiling light) as pictured below.
(instead of writing a proper application for showing shades, I decided to just send different ASCII characters over serial).
I've also tried with and without the lens, and it seems that removing the lens gives better results.