This is introduced in Episode 3 – His Butler, Almighty. I don’t know what he’s wearing beneath the overcoat but it’s likely to be the brown outfit that is shown later in the episode. The coat is probably loose enough that any outfit can be worn under it, really.
This style of overcoat has a second layer over the shoulders. It’s sort of like an attached capelet. It’s supposed to make rain deflect from your body. Technically it’s probably an Ulster coat. This wikipedia page suggests that it could also be a greatcoat with attached cape, although most greatcoats (such as Jack Harkness’ iconic coat in Doctor Who/Torchwood) do not appear to have the cape.
The coat is completely trimmed out in black, including the two side pockets. The cape (and probably the coat beneath) is double-breasted with six silver buttons beneath a classic stand-up collar. From the side here we can see that there is a slit (also trimmed in black) to allow better arm movement. There are four buttons or brads decorating the slit. While the side view does not show the entire bottom of the cape trimmed in black, the front view does show this. It’s probably just a simplification or mistake for that shot. The cape is most likely in two pieces with shoulder seams as there is no back seam either. Also note that there is no slit in the back of the cape.
The sleeves fold up and back, probably permanently. There are three decorative buttons on the sides. Click on the back view picture for a bigger view of this.
The body of the coat is in at least three pieces. In these shots I can see a front piece on each side but no seam down the back. Realistically I think there would be a seam down the back even though we don’t see one. That’s a pretty big piece of fabric to be just one piece. (Also why cut a slit when you could just get two pieces of fabric?) In the manga, by the way, the fabric is a woven houndstooth. See Pinterest board at the bottom for additional references.
Above the slit there are two darts under the little black strap to give the coat some shape. Think of the black strap as being at waist level so the darts shape the back above and below the hips.
Like most of his coats, this one is past the knees but not ankle-length.
He’s got a very cute matching top hat with a wide blue bow. This is also the episode in which he gets the gold-top cane. Details for these accessories (and his other standards – the eyepatch and earrings) are here. The gloves are not on that page; they’re simple black gloves appropriate for a gentleman in the Victorian era.
The boots are very similar to the boots he wore with his burgundy outfit here. I’m not sure if they’re just being drawn differently in different lighting or if they’re different boots. For additional references including a good look at the fabric in the manga, check out the Pinterest board!