If you've written the body of your letter to a state legislature in which, presumably, you articulated your reasons for either supporting or opposing capital punishment, then the introductory section should be relatively simple. As this particular type of letter is a sort-of hybrid letter/essay, the format would differ a little from a typical letter and from a typical essay. In writing an essay or research paper, the introductory paragraph or section should serve merely to declare the author's intent with respect to the remainder of the document. A letter to a politician would begin as follows:
The Honorable XYZ
House of Delegates
Office # [optional]
City, State, Zip Code
Dear Represenative XYZ/Dear Senator XYZ:
The introductory section then begins with the writer's purpose in contacting his or her state representative. A letter to one's elected representative should combine be both professional and personal, a sometimes difficult balance but one attainable. It should be respectful and to-the-point, as elected officials and their staffs are usually very busy and have little inclination to read a letter that is disrespectful, full of typos and grammatical errors, and slow to get to the point. It should address the topic in the opening sentence and then provide a small amount of background in which the writer articulates the reason(s) for deciding to contact his/her elected representative. A rough draft sample could read something like this:
"For some time, I have been concerned about the issue of capital punishment and have followed the debate occurring in the state legislature with some interest. Having listened carefully to both sides in this sensitive debate, and having read varying perspectives, including from victims' families, perpetrators, prosecutors and so-called experts in academia, I have resolved that this state [should eliminate capital punishment/should reinstate capital punishment/should oppose efforts at banning capital punishment]. Allow to me explain my position."
This, as noted, is a rough draft of an introductory section of a hypothetical letter to an elected representative. To reiterate, the student's letter should reflect his or her personal experiences and observations, and should be respectful of the recipient's time and position.