The last time Carter and Sadie saw their uncle Amos was at Sadie's sixth birthday party, when Amos had been among those to witness their argument-inspired birthday cake explosion (at least the icing was tasty). Why had Amos been so distant? Well, the House of Life forbade him from seeing Julius or the kids (6.153). Amos broke the rule in order to try to intervene and persuade Julius not to attempt his second summoning, but, well, we all know how that went.
As soon as the kids need his help, though, Amos is totally there for them, whisking them away to his posh mansion in Brooklyn, feeding them bacon and hot chocolate on Christmas morning, and, um, accidentally betraying them to Set's minions. More on that later.
When we first meet Amos, all we know is that he is rocking a trench coat like nobody's business. Here's how Carter describes him: "He was barrel shaped, with skin the color of roasted coffee. His coat and black pinstriped suit looked expensive. He had long braided hair and wore a black fedora pulled down low over his dark round glasses. He reminded me of a jazz musician" (1.28).
Amos appears impeccably dressed pretty much every time we see him. After being possessed by Set, Amos just isn't himself anymore—and no, it's not because Set is still there, it's because he's broken and depressed about the whole thing. Carter notices, with concern, how different Amos looks post-possession: "He'd lost too much weight. His face looked haggard. Most days he wore his bathrobe and didn't even bother to comb his hair" (41.24).
In other words, Amos's poor physical health reflects his poor mental health. It's another way in which doing the right thing in The Red Pyramid requires a lot of strength and determination. One way we know that Amos is actually a good guy is that he's physically broken up over the bad stuff is happening. Goodness often means having to sacrifice or go through some kind of difficulty in this book.
We can tell that Amos is finally on the mend when he starts to pay attention to his appearance again: "He wore a crisp new chocolate-colored suit with matching coat and fedora. His shoes were shined, his round glasses polished, and his hair freshly braided with amber beads" (41.157). It's almost as though Amos is saying: enough of being a shut-in, I have a whole wardrobe to wear.