Charles Dickens is such a talented writer when it comes to creating unique, memorable, entertaining and engaging characters. His stories are filled with amusing and singular supporting cast members; he makes these supporting characters so memorable through a combination of detailed physical descriptions, giving them unique quirks and mannerisms, and through giving them depth and dimension.
Wemmick has unique physical features, quirks, and depth, and as a result is a recipe for a classic Dickens character. We first meet Wemmick through second-hand accounts, as the infamous Mr. Jaggers rails his pleading clients with the question, "Have you paid Wemmick yet?" We get an impression, from these accounts, that Wemmick is a sidearm to Mr. Jaggers and someone to be dealt with. He is the man that fearful people must pay. When we meet him, Dickens throws in some very detailed and colorful commentary on various aspects of his appearance:
"A square wooden face, whose expression seemed to have been imperfectly chiseled out with a dull-edged chisel...he had glittering eyes--small, keen, and black--and thin mottled lips...his mouth was such a postoffice of a mouth."
He also describes a pock-marked nose, and various rings and tokens on his fingers. So, Dickens characterizes his appearance as a solid wooden block with poorly made dints in it. That is a very unique and interesting image that he creates.
Wemmick also has a soft side, and Dickens rounds out his stoic, stiff, uncaring professional side (that we see as he works for Jaggers) by giving him a quaint, cute house filled with little quirks and an old, ailing parent that he takes care of. Wemmick loves his house, loves his old father, and quite enjoys gardening and other such things. So, he is given depth and softness to his otherwise stern appearance and personna at work.
Overall, Wemmick is a stern, stolid character whose tough side is left behind at work; at home, he is an affectionate, caring, domestic man who cares for family and comforts. I hope that those thoughts help to flesh him out for you! Good luck!