Why did Mr. Whitechapel kill Chapel?

Mr. Whitechapel doesn't actually kill Chapel. He is away from the plantation on the day that the overseer, Sanders Junior, whips the young slave to within an inch of his life. Chapel subsequently contracts a fever after the whipping and dies. The blame for Chapel's death falls squarely on his father, Whitechapel the slave (not to be confused with Mr. Whitechapel, the plantation owner). When Chapel flees the plantation with Mr. Whitechapel's daughter, his own father turns him in, fearing that if Chapel is captured he'd end up being killed.
Whatever Whitechapel's motives may have been in betraying his son's whereabouts, there can be no doubt as to his loyalty towards his master. This is one of the things that separates him from his son. For Chapel has a full realization of the horrors of slavery; that's why he wants to escape up north to start over. Whitechapel, on the other hand, is the quintessential house slave; he believes that the best way for slaves to survive is not to try and escape but to keep their heads down and work hard for their masters:

[A] slave could live a good, long life if he worked hard and presented to his master the most dignified aspect of himself, in order to reciprocate the same manner from that master, the same civility, fairness and even kindness once the relationship grew warm and cordial.