"To What exten is the Friar responsible for Romeo and Juliet's deaths?".
The Friar's actions and involvement did contribute to Romeo and Juliet's deaths. I believe that it wasn't just his actions alone that contributed to the tragedy. The Nurse, Tybalt, "Fate and Fortune", the feuding families of Montague and Capulet and Romeo and Juliet themselves all had parts to play in the tragedy.
The Friar could have done, or not done, a number of things to prevent the deaths of Romeo and Juliet. He should have told the parents about the marriage because he says that " this alliance may so happily prove to turn your households" rancour to pure love." He should have told the parents so that the feud could turn to friendship if his statement was truthful. He should not have given Juliet the potion to make her appear dead. On the other hand Juliet was threatening to kill herself if the problem of her marriage to Paris wasn't solved,so this haste excuses his actions to some extent. When sending word of his plan to Romeo explaining Juliet's "death", he should have at least have told Balthazar about the plan. If he had done this, the unfortunate mix up mightn't have happened. Although these things are valid, I believe very strongly that if he hadn't left the tomb when the guard came Juliet wouldn't have had the chance to kill herself. .
However important these actions were or could have been you can't discount the Nurse's involvement. The Nurse should not have acted as a go-between from the beginning of Romeo and Juliet's relationship. This involvement may have been unavoidable but she at least could have warned Juliet of the consequences of marriage. The Nurse also should have reminded Juliet about the conversation with her mother about the topic of marriage to Paris. After the death of Tybalt and banishment of Romeo the Nurse gave Juliet the bad advice to just go ahead and marry Paris. Juliet takes to this news very badly and in desperation turns to the Friar.