Idiom or phrase associated with a person who blame others for his or her fault?

In a word, such a person is irresponsible. In a phrase, such a person "has no sense of responsibility."

Irresponsible means not responsible.

Responsible is defined by Merriam Webster as:

"2a able to answer for one's conduct and obligations: trustworthy
2b able to choose for oneself between right and wrong"

Webster's New Collegiate Dictionary defines responsibility as: "....Accountability, also moral accountability, as the responsibilities of parenthood; Reliability...."

You might say "If you want the job done, and done right and on time, do not ask X; he is completely irresponsible." Or ".... do not ask X; he has no sense of responsibility."

In looking up definitions for the phrase, I find "awareness of obligations" (e.g., The Free Dictionary). However, as I have always used it, the phrase goes further: the person is not only aware, but cares.

Thus, I like the example quoted in Philiosiblog: "The most important thing is to have a sense of responsibility, commitment, and concern for each of our fellow human beings." Dalai Lama

A "scapegoat" is someone whom someone, or multiple people, wrongly blame for something going wrong. You can use it as a verb too.

(It originates for a primitive ritual in which everyone symbolically piles their sins and problems on a real goat, who was sacrificed to cleanse the community of evil)

To find fault with someone or something. https://idioms.thefreedictionary.com/find+fault