In the play, incest is referred to by Hamlet, and by the ghost, in reference to Claudius and Gertrude marrying. That act, although not technically incest because they were not blood relations, was called incest by both Hamlet and his father as a way to express their distaste for Gertrude's hasty marriage to Claudius. She was related to Claudius, as a sister-in-law, when she was married to Hamlet's dad, so, marrying Claudius, in that sense, could be considered incest. However, most people didn't really think of it that way; the dwelt instead on the positive, the joining of two people in marriage. Because Hamlet is bitter and angry, he calls it incest, as does the ghost of his father.
This motif of incest contributes to the characterization because it shows just how vile Hamlet thinks his mother's marriage is, and some of that distaste is hard to avoid feeling as a reader. We sympathize with Hamlet. We do find it a bit icky and irreverent. Because of this, we are more inclined to dislike Claudius and feel negatively about Gertrude. So, it taints some characters. If she had married some other non-related guy, even if that guy had killed Hamlet's dad, it wouldn't be quite so...disturbing. So, the incest theme puts a negative taint on Gertrude and Claudius.
This negative taint in turn drives the plot; it makes it more suspenseful. Because incest is ringing in Hamlet's ears, he hates Claudius even more. He is viciously cruel to his mother in her bedroom, shouting descriptions at her that are vulgar and disturbing. He feels he can get away with this, considering he feels she has committed the more vulgar crime of incest. So, it adds a lot of suspense and tension, skews certain characters in a negative light, and is an underlying theme in and of itself.