In addition to the superb answer above, you may want to focus on the style of Cash's narrations, his tools, and their relationship to his mother. Cash's duty toward his mother is utilitarian and unemotional: he wants to please his mother with things (making the coffin), and even after she is dead he regards her as a thing, something made, whose function is to be put in its proper place, like one of his tools.
In terms of narrative style, Cash is very mathematic. He makes lists and references angles. The most notable is the bevel, which is a 45-degree edge. Geometry is important in As I Lay Dying: Anse is vertical, always standing (he hates things flat: the road); Addie is horizontal, dead in her coffin. So, Cash is somewhere in the middle: half alive, half dead. In this tragi-comedy, Cash walks a tightrope between the two, nearly dying twice: jumping into the river to save his mother's coffin and suffering from the complications of having his leg set in cement.
Each of Cash's tool represent a member of the family. (See the essay link below for details). Addie is the sawset; Darl is the plane; Dewey Dell is the plumb line; Vardaman is the saw; Anse is the rule; Cash is the square:
In his essay "Faulkner's As I Lay Dying," Tim Poland posits that Cash's “entire identity is located in the fact that he is a carpenter,” and that “[h]is sole means of expression is through his carpenter skills”. Poland notes that when Cash loses his tools to the flooding river, he simultaneously loses consciousness and the ability to speak. Thus, Poland links the loss of Cash’s tools to the “submersion of Cash’s selfhood”. As Poland observes, it is not until the tools are found that Cash regains both consciousness and speech.
Critic Irving Howe in his William Faulkner: A Critical Study, states that the novel, like one by Dostovesky, "stumbles from catastrophe to catastrophe" as the members journey to bury their mother while, at the same time, trying to define themselves. As I Lay Dying delves into the secret life of its jouneyers.
Therefore, one thesis for your character analysis of Cash could be an explanation of his emotional conflicts. As the first child, Cash is the moral head of the family. Howe writes,
Cash, the earnest and amirable carpenter, is the moral head of the family; reflecting Addie's strength and self-possession at the moment she first realizes that 'living was terrible,' he is free of the furies that torment his brothers. Too free, perhaps; his imagination limps behind his conscience, and he is so absorbed in that coffin that he does not notice a family crisis darkening about him.
You may wish to make a statement about these emotional conflicts, and the three points can be his reactions to these emotional conflicts: he does not recognize them, but later become aware of kinship after breaking his leg as his brothers touching him through the torment they cause him, and, taking his place as the objective witness to the family's wreckage.