How should I go about writing an essay on the problem of recycling in my community?

In writing this essay, one of the key factors you should keep in mind is that your grade depends on clearly following the prompt. The assignment has in boldface that the problem you need to address must be one that affects you personally. The problem with the subject you have chosen is that while you can choose to recycle or not, a generic essay about recycling would not actually fulfill the requirements of the assignment. While recycling may help mitigate global climate change or improve the quality of the environment over the long term, that isn't really the point of this assignment; your instructor wants you to focus on something with an immediate impact on your own personal life. 
If you wanted to show how recycling could impact your life, you would need to research waste collection in your community. If a landfill is close to capacity, you might be able to argue that recycling might reduce the need for a new landfill and thus reduce taxes. You could also investigate whether increasing recycling, and including a composting program, could have economic benefits for your community. Another possibility would be looking at the effect of electronics recycling on groundwater quality.
Overall, though, recycling is probably not the best choice of topic in response to this prompt. What your instructor is looking for is probably more an issue such as needing a skateboard park to give teens something to do after school while reducing damage to public structures not designed for skateboarding.