How is Claudius and Gertrude's relationship in Hamlet destructive?

The relationship between Claudius and Gertrude is
destructive in two ways. First, they destroy the potential in their own relationship;
second, they destroy other people literally.

A quick
marriage after a woman is widowed may give rise to suspicions of previous adultery. If
true, deception is not necessarily the best foundation for a positive future
relationship. As their marriage flourishes, it appears that their life's work becomes
putting out the fires of the results of this marriage. Claudius regularly sends Gertrude
to try to speak to Hamlet on their behalf to discover what's wrong. It is in one of
these moments that an accidental destruction occurs when
Hamlet kills Polonius while talking to his mother. Hamlet thought he was killing a
listening Claudius. Now Gertrude and Claudius have to deal with this dead body and
covering up that mess. This weakens their relationship
further.

With Claudius as the killer of Gertrude's previous
husband, and such a close wedding after the death, every audience member jumps to the
conclusion of foul play. Viewers wonder if Gertrude had anything to do with it and they
(like Hamlet) wonder why more time isn't devoted to grieving. These two acts (the murder
and then the marriage) are acts which Claudius and Gertrude contribute to that begin a
destructive process in Prince Hamlet. These acts cause him
to believe the Ghost of his father and they spur him to try to avenge his
death.

Finally, their relationship is fully
destroyed when Claudius accidentally poisons his wife and
Hamlet intentionally kills Claudius.

The relationship
between Claudius and Gertrude is the inciting incident in this drama that ignites the
action of the tragedy that plays out on stage. Without their
destructive relationship, there would be no
play.