Juliet speaks these words in Act 4, scene 3. Friar Lawrence has given her an elixir which should keep her unconscious for forty two hours. He has presented this as a solution to her unfortunate situation. Her father has insisted that she marry Paris that Thursday and she has given in to his request after reconsidering her previously outspoken rejection.

Juliet is, however, already married to Romeo and cannot marry the count, nor does she want to. The potion would put her in a death-like sleep and make others believe that she has passed on. She will then be buried in the family vault and, or so the friar promises, will awake with Romeo ready to whisk her away. Her unsuspecting family would be none the wiser.

Juliet is considering drinking the potion and after considering whether she should involve the nurse, decides that she has to go it alone. In order for her and the friar's plan to work perfectly no one can know what she has done. It is for this reason that she decides, "My dismal scene I must act alone."

She states, "I must perform this unfortunate act by myself." If the nurse should know, she might, in an unguarded moment, divulge the truth. Juliet is prepared to take this desperate step, although she muses that the concoction might not work and that the friar may have some ulterior motive in giving her the potion. If she is dead the friar might get away with the fact that he had married her and Romeo already. Furthermore, she might suffocate and die before Romeo even arrives.

Juliet further thinks about how terrible it would be when she is surrounded by rotting corpses and to be in a dark chamber next to Tybalt's bloodied corpse. She does, however, drink the potion despite her many fears.  

She means that she has no choice but to make her own decision and take her life (literally) into her own hands.

Juliet says 'My dismal scene I needs must act alone.' or in other words, 'I must act my dreadful scene alone.' This is said as Juliet prepares to drink the potion.