How do I go about framing a half wall into a full wall?
If you were to just frame up the current opening above the half wall, you're going to run in to problems with the wall flexing, possibly cracking the drywall.

Therefore, at a minimum, I would put at least a doubled-up stud at the end of the half-wall that runs from the floor to the ceiling. (But first remove drywall and trim from the end and top of the half-wall.)  The floor-to-ceiling double stud will resist the flexing. Then you just fill in the opening with a header, footer and studs; hang some drywall; and patch and paint.

However, to do it "right" you would remove the half-wall and just frame in a new wall using traditional building methods.

Personally, I would choose the right way. It's probably not much difference in effort and price and you end up with a more standard finished product.