The answer, in reality, is difficult to define because DuPont himself never gave a motive other than insanity. Given his erratic nature in the months leading up to the shooting, it was likely alcoholism and paranoia from cocaine psychosis that drove him to shoot Dave.
In the film, however, much of the plot is spent building motives. When DuPont starts the wrestling program, he is clearly a dilettante, using his considerable resources to cultivate wrestlers out of misguided patriotism, self-indulgence, and self-glorification. In Mark, he sees a kindred spirit: Mark was largely raised by his brother Dave, trained by him, and apparently incapable of escaping his shadow. DuPont was also without a father at an early age, and is unable to escape the shadow of his family's cultural identity. Their conversation in the first third of the film points to Mark seeking a surrogate father, and DuPont, having realized his only childhood friend was as a result of his mother paying the groundskeeper for his obedience, desperately wants to be important to someone.
When Mark first introduces DuPont to Dave in his hotel room, DuPont sees the he has a wife and children who love him, and he is happy. It is clear in this interaction that DuPont is uncomfortable, and it can be supposed that DuPont's inability to understand Dave's happiness is both irksome and confusing for him. This is examined further when DuPont visits Dave's family at his house on Foxcatcher in an attempt to integrate himself in Dave's life. Being politely rebuffed by Dave reflects DuPont's mother's cold indifference to him; he is lonely, unloved, and incapable of buying or sustaining a family life.
When DuPont leaves the hotel room, Mark takes up for him in his absence, driving a wedge between Mark and Dave. Later, DuPont's mother intimates that wrestling is beneath him, leaving him deeply wounded. Having felt that the wrestling team would impress her enough to validate or escape the shadow of his heritage, he takes out his frustration by exploding on Mark, hiring Dave 'no matter the cost', thus indicating that his brother is a better man. If DuPont, who can be just as cold as his mother, can't escape his shadow, he feels Mark shouldn't escape his either. Later, DuPont attempts to impress his mother by showing off with the wrestlers, but she sees through this simple facade and blatantly dismisses him, further destroying any hope of DuPont validating his indulgence.
When Dave arrives at Foxcatcher, DuPont realizes that he destroyed his relationship with Mark, and throughout the rest of the film, he clearly wishes to repair that bond; this is understood when DuPont attempts to observe and encourage Mark's workouts, join in with his radical weight loss following his eating binge, and insist on being in Mark's corner during the 1988 Olympics. During this time, DuPont sees Mark and Dave drawing closer together and cannot understand how their familial bond was so easily repaired. When Dave insists that Mark cannot stay at Foxcatcher, DuPont is crushed to see that Mark hates him, and that he may have incidentally encouraged Mark's reconciliation with Dave. He can't even look at either of them when Dave negotiates for Mark to leave.
Finally, DuPont views the fluff documentary he commissioned to ensure him that his self-glorification was validated, and towards the documentary's conclusion, Mark embraces DuPont after 1987 World Championship victory, giving DuPont the closest thing to validation he's ever felt. Reflecting on their friendship, his coldness toward Mark, Mark's repaired bond with Dave, and the fact that he pushed Mark away by inviting Dave to Foxcatcher, he decides that he has only one course of action: kill Dave. This decision could either be predicated on destroying Mark and Dave's bond, eliminating Dave so that Mark can be his own man, or both. Either way, DuPont sees removing Dave as paramount to repairing his bond with Mark.
When DuPont drives up to Dave's estate and asks if Dave has 'a problem' with him, he refers to the fact that Dave never let him into his family, echoing the same betrayal he felt when he found Mark goofing off, and when he saw Dave and Mark shutting him out. In much the same way he exploded on Mark, he shoots Dave and drives away, perhaps oblivious to the repercussions of his actions; his wealth has given him a lifetime of isolation and indulgence, forever trapping him in the illusion that his excesses, no matter what the end, were beyond reproach.