Word for someone who's incompetent

"underachiever", "boat anchor", "doorstop" all have varying degrees of negativity and rudeness but what might also play a role in your choice of word, of course, is whether this person's performance has any affect on you and the class. If the person is just not trying and is lazy, as slacker implies, a derogatory word such as "gold bricker", "dead-beat", "goof-off", "loafer", "slouch", "sloth", "back-slider" or "lazybones" could also work. (per www.thesaurus.com)

What about deadbeat, idler or loafer

Slacker implies laziness, so that may or may not be a good word for what you want. If this is a matter of ability rather than motivation, then I like Kris' suggestion of incompetent instead.

Another word you might consider is inept. The words unskilled or bumbling might work, too, but they suggest a shortcoming in dexterity more than intelligence (such words might work well in a trade school, though, for the situation you describe). If you want to emphasize results (or a lack thereof), you might try unproductive.

Yet another way to express this sentiment is to forgo the synonym, and instead opt for one of the many catch phrases used to humorously describe the shortcomings of individuals in relation to their peers, such as:

Not the sharpest knife in the drawer.
Not the sharpest tack on the board.
Not the sharpest tool in the shed.
Not the brightest crayon in the box.
Not the brightest bulb on the tree.

although some of these might be considered trite.

You might say “He was hindmost in the group.” Hindmost means the most to the rear. Or say “On a good day, he was last in his class.”

If you want to laud a person's effort, even if he fared badly in a course, you might say something like “In spite of being worst-prepared and most-disadvantaged at the outset of the course, and in spite of always faring last, he stuck with it and was not among the many who flunked out.”

You could say:

He was the dunce of the group.

dunce:

a slow-witted or stupid person