I don't have chemistry in my background (or I don't have much physics, or meteorology...). Is that a problem?

Not necessarily. Atmospheric chemistry is an interdisciplinary science. My graduate students come from a diversity of backgrounds from applied math to engineering to chemistry and everything in between. Some students come in with zero chemistry; others come in with little physics. Many have had little or no exposure to atmospheric science, while others have had quite a bit. Gaps are expected, and the purpose of your courses at Harvard will be to fill these gaps. I do expect applicants to have had at least two years of college math.