Is a criminal who suffers from a mental illness or personality disorder really at fault for the crime(s) they commit?

It depends: This isn't exactly a medical question, it's more of a legal or moral one. But laws in the us generally hold that if a person cannot tell right from wrong due to a mental illness, then they are "not guilty by reason of insanity" (or similar language depending on the state). Most mental illness is not severe enough to meet this standard, and the "insanity defense" is actually used fairly rarely.