I downloaded and installed the Modern King font (http://www.dafont.com/modernking.font). The $ character is supposed to display an O with a crown over it. But when I actually use the font, it does not display the entire character. All it shows is the O with the bottom left and right corners of the font. Is this a problem with the font itself, or is it a problem on my end that can be remedied?

The problem is with the font. Or rather with the maker of the font who did not have any idea what he/she was doing putting the crown over the O.
In short, very badly made and unusable. Same old ascender descender problem.

The only solution is to contact the author to he correct his work.
http://www.behance.net/rawadsaghir

Whatever way this is "corrected", the result is a ridiculous line gap. That crown should ever have been added as is.