And I doubt the Emperor does his own hiring, so Luke could become some petty officer and never be found out for being Darth Vader's son.
If Luke graduated from the Academy, he'd be an Ensign in the Starfleet, or some kind of Lieutenant or Pilot Officer in the TIE Corps. Not a petty officer. (Realistically, in ANH I'd peg him for a brief career as a junior enlisted man followed by a return to farming. But I guess the Imperial Academy isn't terribly selective when it comes to its flight cadets.) Saxton's take on the Imperial rank structure is here.
As for being found out, I'm sure officers undergo at least some background investigation, as well as a genetic screening. Those would've set up all kinds of red flags. Assume a four year course at the Academy, followed by one year of flight instruction, then five years of active commissioned service, that's ten years for someone to notice
a) This guy shares half his DNA with Darth Vader,
b) He has the same name as Darth Vader (or the Jedi Councillor who liased with Chancellor Palpatine, depending on who's doing the research), and
c) He's Force-sensitive.
One could argue whether or not Luke would last that long in service to the Empire he claimed to hate, but I think even applying to the Academy would've brought the Empire down on Tatooine in force.