How can I write a letter to the magistrate in Trifles pleading that Minnie is not guilty?

To make such a letter look official, try looking for templates of magistrate letters. This is as simple as googling the key words: template, letter, and magistrate. The letter would begin with something like "Dear Sir:" and a few paragraphs would follow. The letter should also close with something formal such as "Yours Sincerely, (Insert Name)." 

In the bulk of the letter, you state your case. Are you arguing that Minnie did not kill her husband? If that's the case, you would have to present evidence and reasoning that exonerates Minnie. You might add that her stoic or flippant manner when Hale discovers her is shock. It's the result of finding her husband dead. You might also argue that the dead bird could have been killed by an intruder, the same intruder who killed Mr. Wright. 

If you are going to argue that Minnie had been justified in killing her husband, build the case around the idea that Mr. Wright (ironically named) was calculatingly bad, and perhaps abusive, husband. We can only speculate about physical abuse because it is not overtly mentioned. But Mrs. Hale and Mrs. Peters can testify to the fact that Minnie was much happier and outgoing before her marriage. They note John was a "hard man" to live with. And Mrs. Hale adds " . . . I don't think a place'd be any cheerfuller for John Wright's being in it." One could argue that moving Minnie to the farmhouse was like a type of imprisonment. Given Minnie's love of singing, the supposition that John killed the bird is a good piece of evidence. One might argue that this was the last straw for Minnie. After a marriage of loneliness and isolation, killing the bird set Minnie over the edge. If a jury could not justify the murder, they might be able to find that John forced Minnie to a psychological break and this led to her retaliation.