Although the vocabulary used is quite rich, I don't think the majority of words and phrases used are so archaic that they require a "modern translation". Even phrases that are not used today, like "your letter of the 18th Instant" are not too difficult to understand.

Take a few passes over the letters with a dictionary to help you clarify the meaning of individual sentences. Then go back and see how they progress in the argument.

This is all about the art of the subtle (and not-so-subtle) insult. Hamilton is a gifted writer, and doesn't resort to a crude "F.U." or one-line "yo-mama" barb. I find his letter of June 20 to be hilarious, and attempting to summarize or simplify it would detract from its beauty - like asking an accomplished ballet dancer to move with less grace.