I'd run some Internet searches for images of animals with claws, to get an idea of what the real thing looks like. Remember that the Gillman was completely made up, so there's no real right or wrong, necessarily. You might want to look for images from the movie, too, though since it was in black and white, the pictures might not help too much. From stills that I remember from the movie, the color of his claws isn't too obvious.
I've painted powder horns on my 54mm 18th century figures, and what I've done is to look at real cow horn, then used 2 colors, a brown for the base and tip of the horn, and a cream color for the main color of the main part of the horn. I painted the cream first, then added the brown at the base and tip, and then used a brush with a square edge to drag the brown into the cream color, without blending the two. I did the same thing at the tip. I use gloss enamels, by the way, Testor's in the little square bottles, but oils would work the same way, because of their viscosity and slow drying time. I don't think you could do the same thing with acrylics.
Hope that helps, best regards,
Brad