Eliza is a free black woman who works alongside Mattie's mother Lucinda in running the coffeeshop. Nathaniel is a young apprenctice to the painter Charles Wilson Peale who has a crush on Mattie. Aside from a casual acquaintance, Eliza and Nathaniel really do not interact very much in the story; they have no relationship to speak of. I wonder if you meant to ask about the relationship between Mattie and Nathaniel instead?
At the beginning of the book, Mattie and Nathaniel are about the same age, friends who might be interested in developing more of a relationship as time goes on. Mattie recalls how Nathaniel "had rubbed snow in (her) face and chased (her) across the ice", a sure sign of interest; later, they go to see the historical launching of Jean Pierre Blanchard's hot air balloon together, and Nathaniel gives Mattie flowers during the height of the fever epidemic. Mattie's mother does not approve of Nathaniel in the pre-epidemic days, snobbishly calling him unsuitable, with "no future...a scamp, possibly even a scoundrel" (Chapter 5). After the fever, however, Mattie has grown up a lot and her mother's influence is much diminished. Mattie says they have "an understanding", indicating that when the time is right, their relationship will most probably develop into something deeper and more permanent (Chapter 28).