This is, of course, a matter of personal opinion.  I will look at the president that I think did the worst job of abiding by Article II of the Constitution.  I will say that John Adams did the worst job.  The main reason I say this is because he allowed the Alien and Sedition Acts to be passed and enforced even though they were perhaps the most blatantly unconstitutional laws ever passed by our Congress.

There are two things that Article II does that make me think that Adams did not do a good job.  First, Article II says that the President must “take care that the laws be faithfully execute take care that the laws be faithfully executed” and it says that he (or someday she) must take an oath in which they swear that they

will faithfully execute the office of President of the United States, and will to the best of my ability, preserve, protect and defend the Constitution of the United States.

I would argue that Adams failed badly to do these things when he allowed the Alien and Sedition Acts to be passed and implemented.

The Alien and Sedition Acts were terribly unconstitutional.  Among other things, they essentially made it illegal for anyone to criticize the government.  The laws were passed because the Federalist Party wanted to keep power and because they felt that the ideas of the Democratic-Republican Party were bad for the country.  Therefore, they passed these laws that made it very difficult for the Democratic-Republicans to express their opinions.  This is a terrible violation of the right to free speech as enshrined in the First Amendment.  Because Adams failed to protect and defend the Constitution, and because he failed to faithfully execute the laws (the First Amendment), I would say that he did a terrible job of respecting his job as outlined in Article II.