How has Huck's view of Jim changed throughout the novel? In what ways has their relationship changed, grown, and shown a new maturity?

When Jim is introduced, it involves a prank played on him while napping, by Tom Sawyer—the ringleader—and Huck. Huck is a passive observer during the practical joke, and in the subsequent foolishness: Jim is convinced that “witches bewitched him and put him in a trance.” “Jim was most ruined for a servant, because he got so stuck up on account of having seen the devil and being ridden by witches.” At this point in their interactions, Jim is looked upon patronizingly, as comic relief.
Huck later resolves to “take to the woods...and tramp across the country,” and he makes a tolerably good go of it. He’s satisfied, smoking fish over a fire, but realizes he’s lonely, when he comes across Jim, also in hiding. It’s evident that, each for their own reasons, they’ve gone underground. As two marginal souls, and misfits, each estranged from society, their perceived status difference, i.e. the presumption of racial superiority or inferiority, is blurred. But only blurred, not cleared away. There’s no narrative time-compression; Huck’s acceptance of Jim as fully human evolves slowly.
As they travel the river, Huck still looks askance as Jim dreams about what he can do once they’ve escaped and he becomes a free man. Social conditioning—thinking less of the Other—is, and seems that it always will be, too entrenched a custom to yield readily.
But the culture from which they make their escape has it’s own particular values. Huck has a moral dilemma. Is he helping a worthy man to his freedom or committing the theft of officially-owned property? Huck does what he feels to be right, as his relative understanding and empathy for Jim develops: if it’s a sin to help a slave, he resigns himself to Hell.

Huck's view of Jim undergoes quite a considerable change over time. Initially, Huck, in common with virtually all white people at the time, regards Jim as little more than an ignorant slave. Observe how Huck plays a number of cruel pranks on Jim, which make fun of his many superstitions. Probably the worst example of this is when Huck gaslights Jim into believing that a thick fog which had separated them never actually happened.
Yet over the course of their adventures, Huck and Jim draw closer together, forming an unbreakable bond of friendship. The numerous scrapes they get into make Huck see Jim not just as a good friend, but as a true human being deserving of care and respect. Not only that, but Huck finally sees past Jim's race and accepts him as his equal.