You do not need another Bachelor's degree. After graduating with a bachelors degree in natural science, I taught high school for three years, and then went to graduate school to study air quality in an environmental engineering program. Many students in my group studied air quality / climate change interactions and climate change was a significant part of our class curriculum. If you have good computer skills and a good academic record, then you should have no problem finding a university that will fund you to do Master's work in climate-related environmental sciences. Universities need graduate students because they perform a fundamental duty (e.g. teaching or researching) at a low cost compared to a professor or post-doc. However, you should keep in mind that after you graduate, you may not be able to find a job doing what you want in climate-science, unless you are prepared to write funding proposals.
Climate change is a vast field which is not all about the atmosphere. Biology plays a significant role. I would suggest you try to get involved with climate-related marine biology, changes in ecosystems due to climate change, or the effects that increased carbon/temperature have on plants/animals. Though, jumping into atmospheric science as your primary area of study will put you at a competitive disadvantage since many graduate students will have studied that in their Bachelor's program.
As for a book, I suggest you read David MacKay's "Sustainable Energy, without the hot air", and I also suggest you familiarize yourself with the IPCC reports and Global Carbon Project.