C/N I've always read in was the person person who doesn't care about the means or the ends, just that the result benefits me or my accomplices

Neutrality is more like doing what it takes to achieve a cosmic state. It places no value on ones self or welfare of others. What you're describing is Chaotic Good.

Chaotic Neutral just means that there are no rules for how to achieve this state. One choice is as good as any other. The more rules, the more Lawful.

C/E would be the sort that is exemplified by the social climbing boot licker

This one might be right depending on how you envision this character. There are certain rules to high society and if their actions are constrained by any of them then they can't be Chaotic. If they only have like... one rule then maybe they're Neutral.