Christopher's mother left Christopher and her husband to live with her lover, the neighbor across the street.
She, thus, has abandoned her son and her family. On a number of occasions she admits or Haddon infers that Mr. Boone is a more patient, more dedicated parent, at least to Christopher's needs.
No, the mother isn't dead. She's living in London. While she cares for her son's well-being, she cares too little. One could argue that the father withholds her letters from Christopher because, 1. he's still angry and bitter at her abandoning the family, and 2. he didn't want to confuse Christopher. It suggests that by keeping the letters (not tossing them!) he will - one day- share them with an older Christopher who will be able to understand the meaning of his father's behavior and his mother's actions toward him and his father.
Let us remember that the mother of Christopher is a character that we actually never meet, as she died prior to the beginning of the novel. However, that she loved her son is absolutely clear, as the discovery of the letters that she wrote testifies. Although Christopher never visited her when she was in hospital, this in no way indicates that they did not enjoy a close relationship. Consider the following quote from the novel where Christopher describes himself and explicitly relates that description to his mother, showing that although she is physically absent, she is very much still a part of his life and thinking:
I do not tell lies. Mother used to say that this was because I was a good person. But it is not because I am a good person. It is because I can't tell lies.
We can see here how Christopher has such a strong literal definition of the truth, and this is balanced with his mother's love for him that causes her to think better of him than he actually is. Such quotes allow us to infer that she deeply cared for her son and was a very empathetic individual.