Jonathan looks like the famous horror actor Boris Karloff because he has had plastic surgery to disguise his appearance as he is wanted by the police. His surgeon was drunk while performing surgery and had Karloff on his mind while working.
When people mention the resemblance, Jonathan becomes enraged that he would be compared to someone who is known for portraying monsters in the movies.
The irony, of course, is that Jonathan is a murderer many times over, and is, therefore, a monster himself. He is such a psychopath, he has no clue that he is such a fiend, but feels his actions were warranted with each killing, and plans to kill his brother Mortimer as well.