This page of the Witcher 3 Wild Hunt guide contains a walkthrough for The Play's The Thing story quest. The solution below will help with preparations, and shows whether choice of the actors affects the outcome of the quest.
Talk to Priscilla in the inn. You must prepare a play to be shown in order to get the attention of a known doppler. This quest has few phases. First, during the conversation with the poet, you decide how the play should go. After that Geralt doesn't have much to say and it doesn't matter anyway.
When the play will be ready, go to Irina to prepare the show. You must buy a ticket to walk inside (50 crowns). After the conversation with Irina, you must continue your preparations: find security guards and people that will make the play known.
People responsible for advertising are a group called the Puffins. Talk to them in their quarters. They will agree to help you if you take care of the bandits lurking nearby. You can attack them (you must only kill one - level 11), bribe them for 50 crowns or use Axii sign.
Next find security in the docks. You will see fighting men - this is your target. You can pay them 70 crowns upfront or try to fight two enemies at once. If you win, they will help you for free. However, if you lose, you will have to pay twice the amount.
Finishing the quest
Go back to Irina. Decide who will play as whom. It won't matter too much. After the play and a short conversation, you will receive about 100-150 crowns and a new main quest - A Poet Under Pressure.