What are some different signs of Jim and Huck's relationship growing throughout the whole novel?
Pay attention to the way Huck refers to Jim and to his reactions to Jim's increasing role as a father figure to Huck. For example, in the beginning parts of the novel, Huck refers to Jim as "Ms. Watson's nigger Jim". The use of the possessive here indicates that Huck views Jim as nothing more than property. Barely human, in fact.
As the novel progresses and Jim begins to subtly influence the moral development of Huck's character we see this demonstrated in Huck's reaction. In the scene where the two are separated by the fog and Huck plays a trick on Jim, Jim reacts harshly, calling Huck "trash" (which was more of a racial epithet at the time than "nigger" actually). We see the development of Huck here when he goes and apologizes to Jim, showing remorse for his joke. Obviously Jim's opinion of Huck matters to Huck.
Further on in the novel, we see perhaps the most telling sign of the development of Jim and Huck's relationship when Huck decides to write the letter to Ms. Watson, telling her where her escaped slave is. After writing the letter, Huck cannot send it. In fact, he goes the opposite direction. He tears up the letter saying, "All right then, I'll go to hell." The fact that Huck is choosing to preserve Jim's freedom over saving his own soul (!) is a powerful commentary on the influence Jim has had over Huck, causing him to become a young man who sees Jim and his freedom as more important. When someone can willingly sacrifice himself for the good of another, he is truly a man.
Hope this helps!