After reading Matt Ridley, Richard Dawkins, etc. I wanted to know more about what was really happening inside my cells. The popular science bookshelf stops there, so you need to go to the undergraduate textbook section instead. Textbooks are not normally read cover to cover, but I found this one to be perfectly readable.
Assuming almost no prior knowledge, Lesk describes nucleotides, DNA, RNA, amino acids, replication, gene expression, epigenetics, cell structure, DNA sequencing, HOX genes, and much more. He also touches on the major DNA databases and algorithms (although he has another book specifically about bioinformatics).
No prior knowledge is necessary, although it might help to read a few lighter books first. This is a fascinating insight into the complexity of life. If you are interested in how DNA works, or what (probably) causes cancer, then I strongly recommend this book.