In Chapter 17 & 18 does John's contemplation of death conflict with our impression of his strong character in Brave New World?
I am not sure that I would characterize John as a "strong" character, although I'm not exactly sure what that means. John was an "alien" being in Brave New World; he had a set of values that were in conflict with almost everything he saw in their society. His reactions were passionate, but I think they came from his inability to understand their world and his holding on to what had made sense for him throughout his life.
Toward the end, he seems to be overwhelmed by his inability to deal with or change their world, so he opts for suicide. I also suspect that this was a "practical" need in the novel. All the characters had served their "purpose," so Huxley may have decided to wrap up the lose ends by having the Savage end his life and just let us imagine that things continued in BNW has they had "always" been. In a sense this reenforces the theme that the individual has little weight in their society, no matter how much Shakespeare he/she knows.
I would say that it does not.
In Chapter 18, John the Savage exiles himself because he cannot stand civilization. Then, when civilization comes after him, he ends up killing himself. He does this after he gets so caught up in the heat of the moment that he participates in an orgy.
I do not think that this clashes with the idea that he has a strong character. I think it shows that he realizes that there is nowhere in the brave new world that a man with his values and ideals can go. If he cannot live according to his values, he would rather die. I think that is more a sign of strength than of weakness.