How do Mr. Darcy's first and second proposals show his transformation in character and disposition in Pride and Prejudice?
1. a person's usual temperament or frame of mind
2. a natural or acquired tendency, inclination, or habit in a person or thing (Collins Dictionary)
Even if the first definition of "disposition" is being used, it may be difficult to argue that there is a change to Darcy's disposition; it may be clearer to argue that there is no change in Darcy's disposition. In other words, there is no change in his natural, usual inclination, temperament or frame of mind. This is confirmed by Elizabeth herself when she expresses the opinion in her musings that, now that she was never to see him again, Darcy was a man of just the sort of disposition that could make her happy in marriage. It is also confirmed by her remark to Wickham, as her new brother-in-law, that Darcy's disposition had not improved but that her knowledge of his disposition had improved.
She began now to comprehend that he was exactly the man who, in disposition and talents, would most suit her.
"When I said that he improved on acquaintance, I did not mean that his mind or his manners were in a state of improvement, but that, from knowing him better, his disposition was better understood."
Nonetheless, something has changed about Darcy. That something is best described in relation to the text as "character" or "characterization" and in relation to the persona of Darcy as deportment and attitude. He says of himself that he had never been taught to regulate his selfish practices or his mood, though he was not selfish by "principle," and was "spoilt" by his parents and left to follow his good principles in "pride and conceit" and to be "selfish and overbearing." Elizabeth's rejection of his first proposal "humbled" him and replaced his pride and conceit with a kindlier deportment that valued other people, like Mr. and Mrs. Gardner, as of equal worth as himself.
Darcy's first proposal illustrates the conceit and pride that he attributes to himself after he and Elizabeth have reconciled their love for each other. His first proposal was indeed a proposal with not only an expression of his feelings for her and of what he offered her materially but also with a request that she be his wife. His second proposal, which wasn't in fact a proposal but a mutual agreement that their feelings were in accord, illustrates how thoroughly Elizabeth's rejection of his first proposal of marriage had humbled him and taught him to value other people as having the same worth as he had: we see Darcy without conceit and pride and without selfishness.
" ... But your family owe me nothing. Much as I respect them, I believe I thought only of you."
Elizabeth was too much embarrassed to say a word. After a short pause, her companion added, "You are too generous to trifle with me. If your feelings are still what they were last April, tell me so at once. My affections and wishes are unchanged, but one word from you will silence me on this subject for ever."
Elizabeth, feeling all the more than common awkwardness and anxiety of his situation, now forced herself to speak; and immediately, ... gave him to understand that her sentiments had undergone so material a change, ... as to make her receive with gratitude and pleasure his present assurances.