In Act 4 Sc. 7 of Hamlet, how does Claudius' backup plan to poison Hamlet's drink develop the plot of the play?
Act IV is typically the act which shows the quickly occurring consequences of the climactic action in Act III. Thus, Act IV is usually called that of falling action. In this case, that action was killing Polonius. One of the consequences in Act IV is Claudius' fear of Hamlet which prompts him to plot his murder. Hamlet has survived the first attempt which has made Claudius even more desperate. Now, he has plotted with Laertes, Polonius' son, to have Hamlet killed "accidentally" during a fencing match. However, the initial ploy to poison Laertes' foil will only work if Laertes succeeds on making a viable hit upon Hamlet. Doubting Laertes ability, Claudius has developed a back up plan which involves coercing Hamlet into drinking some poisoned wine at some point during the match to celebrate his skills.
However, the consequences in a tragedy are not confined only to one character. As the play resolves in Act V, the reader finds that virtually all characters are killed by this plan, both innocent and not.