This next letter is the one that I believe is the most important regarding uncovering the relationship of Hamilton and Laurens.

In spite of Schuylers black eyes, I have still a part for the public and another for you;

-This translates to "I find Eliza beautiful and love her but I have similar feelings for both the people and for you". This on its own shows a relationship more than just your average friends. Hamilton is saying that no matter how beautiful Eliza is, part of him feels "things" for Laurens.

so your impatience to have me married is misplaced; a strange cure by the way,

-The question to be asked here is, a cure for what? Based on its position in this letter as well as the way it follows the previous line suggests it may be a cure for the "feelings" hinted at. As being Attracted to the same gender would be considered an illness this "cure" could quite possibly be referring to that. This possibility is even more likely when taking the line before into consideration which notes that the cure is marrying a woman.

as if after matrimony I was to be less devoted than I am now.

-Here Hamilton makes it very clear to John Laurens that the "cure" didn't make him any less devoted and that the cure had not worked. This mean that failed cure = Hamilton still devoted to Laurens.

Let me tell you, that I intend to restore the empire of Hymen and that Cupid is to be his prime Minister. I wish you were at liberty to transgress the bounds of Pensylvania. I would invite you after the fall to Albany to be witness to the final consummation.

-This has two very different meanings. 1. Laurens is invited to the wedding
2. Laurens is invited to a threesome
The intention of this is not made clear.

My Mistress is a good girl, and already loves you because I have told her you are a clever fellow and my friend; but mind, she loves you a l’americaine not a la françoise.

-This is basically Hamilton clarifying that Eliza loves him platonically and not romantically or sexually. If the intention of the previous section was to invite Laurens to a threesome this would mean Laurens wasn't invited for Eliza as she does not feel that way, which would make the question, who was he invited for?

Adieu, be happy, and let friendship between us be more than a name

A Hamilton

-This is by far the letter that most suggests a relationship. The "cure" mentioned is fairly likely to refer to homosexual feelings and relations which in turn makes at least Hamilton very likely to be bisexual (as he notes that he also loves his wife and that he is attracted to her).