The relationship is something of a strange one. Based on the way Hamlet describes it and the way his father's ghosts relates his betrayal, it appears that this is a passionate and lustful relationship, and yet this is not exactly supported by the characters' actions. The couple gets married shortly after the death of Hamlet's father. This is certainly strange behavior and a sign of disrespect to Hamlet's father. Some might suggest that the two had even been engaging in a sexual relationship before the death of Hamlet's father. This could possibly be one of several motivations for the king's murder.
This does not appear to be the case, however. The play shows several signs (such as Hamlet's play, and Gertrude's own actions) that Gertrude was unaware of the plot against her former husband. While this does not mean a relationship did not occur, it at least removes the possibility that the two planned the treachery together. It seems more likely that Gertrude's role in the play is somewhat passive. She naively closes her eyes to the crimes of her new husband and tries to pacify her son to hold everything together. She remarries quickly, not because she is so in love with Claudius, but because of a desire to keep things in order.
After a lifetime of obeying her royal husband she appears simply to allow Claudius to fill the vacant role in her life. She trusts him to help her discover why her son is acting so strangely, and she hopes that Hamlet will view him as a father figure. There is little sign of any lustful passion on her part, and she even attempts to save Hamlet at the end of the play. For this reason, it seems that while Claudius might view marrying Gertrude as a part of his treachery, Gertrude is merely passively trying to continue life as it was before. She wants things to remain as they were and for Claudius to continue in the role her husband left behind.