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So basically Burr is mad at Alexander, but he can't ruin his chances at success by doing anything to hurt him.
Then he has an idea; get somebody else to do it!
His boyfriend George Eacker eagerly agrees to do it for him, publicly talking badly about Alex and his affair.
Later, the pair hear news that Philip Hamilton is mad at Eacker. Burr tells him to agree to a duel of the matter comes up, which it obviously will as the boy has a temper as short as his father's.
They intend only to hurt him, so that Alex will be so distracted taking care of his injured son that he won't interfere with Burr.
However, they didn't count of infection, and Philip died.
Ah, well. These things happen! At least that got Alex out of the way for good.