Apply grease paint. This is something that is not a common item anymore, but was popular around the stage in the early part of the 20th century. Believe it or not, Groucho's mustache was actually grease paint. While he did from time to time grow one, more often then not he applied it. So, taking your finger, apply a thick rectangular shape of grease paint on your upper lip. Do the same thing with your eyebrows, to enhance that thick, bushy look.
Part your hair in the middle. Groucho used to part his hair straight down the middle. Use pomade and a comb to do this. Make sure not to "slick" the hair down to the side, but merely part it. The hair should not look wet, but simply flat and separated.
Get into the clothes. Groucho was famous for wearing long tails and slightly hunching over as he walked. So once you put the tux on, bend slightly at the waist as if you were leaning forward.
Add the glasses and trademark cigar. You can get Groucho glasses (some attached with mustaches) at many costume stores.