What is the most important aspect of plumbing door frame
There's a built-in gap in the height and width of the opening. so trimmer/king plumb parallel to the wall's length isn't crucial (to a degree). It's far more critical to get the wall itself plumb perpendicular to the rough opening. Nearly everything else can be shimmed out later, assuming reasonably accurate dimensions. 

Using a level of adequate length (to account for curves in the studs--I'd consider 4' to be a minimum), check plumb on the narrow edge of the king studs. How precise you get is up to your personal standards. A centered bubble indicates plumb or level. If it's not centered, it's not plumb or level. Flip the level end-for-end or rotate it 180 degrees and compare readings to gauge level calibration.

If the wall isn't plumb and flat, you have two possible problems:


The door will swing one way or the other due to gravity.
The R.O. will have a twist, requiring some creative jamb mounting and casing installation to allow proper door operation and get it looking right.


If all the walls on the project are reasonably plumb and straight (top and bottom) there's no reason that your openings won't be as well. 

You mention king and jack (trimmer) flush. That's important to keep the wall at the correct thickness. A thickened wall makes trim work more challenging, but won't typically affect the installation or operation of the door itself.