Mr Wickham has a number of reasons for accusing Mr Darcy of disgracing his father's memory. Firstly, and most importantly to him, he did not allow Wickham to take up the living that was promised him by the old Mr Darcy, instead, according to Wickham, giving it to another man, and secondly, by not providing for Wickham in the way that Wickham feels the old Mr Darcy would have. Note what Wickham says about Darcy's father:
He was my godfather, and excessively attached to me. I cannot do justice to his kindness. He meant to provide for me amply, and thought he had done it; but when the living fell, it was given elsewhere.
Darcy therefore dishonours his father's name by not upholding what the old Mr Darcy promised Wickham during his lifetime. By not fulfilling his father's wishes, he shows himself to be an unworthy successor of the old Mr Darcy's generosity of spirit and kindness. Interestingly, Wickham is somewhat vague about the reasons why Darcy did not fulfil his father's wishes, stating that he chose to ignore it because it was an informal bequest. Of course, later on in the novel, the real truth becomes clear and the reader--and Lizzie--realise that this was not the case at all and that Wickham willingly refused the living and was paid off in full.