I've got students at my uni who have been taught "A level" chemistry by biochemistry graduates that are previous alumni of my university who have gone into teaching. We've also got students who have been taught biology again by biochemistry graduates. This is not an unusual scenario nowadays thanks to the flexibility and multi-skilling demanded of many secondary school teachers. All the same, it should enable you to see that biochemists have a deep understanding of both chemistry and biology.
Yes, biochemistry is very interesting and the science allows you to mesh together both chemical and biological processes in a way that's quite unique in science.
Although I'm a professor at a leading UK uni and a visiting professor at a number of others including a medical school, I'm also an adviser for the government and work at one or two of their research agencies for part of my time. Now and again, I've even been known to edit or write the odd textbook. Believe me, it's never a dull moment and if you're really interested in biochemistry, go for it.