Note: All text in block quotes (like this sentence, for example) are the Kli Yakar's actual words in translation. Words in block parentheses [like this!] are my additions for clarity or to provide the original Hebrew term used by the Kli Yakar.
The Kli Yakar states that this parsha hints at three things:
- Lashon hara (forbidden speech)
- Haughtiness (which he terms gasut haruach)
- Envy (which he terms tzarut ayin)
He then explains that haughtiness and envy come from the same place, and both lead to coveting money and outright stealing.
According to the Kli Yakar, these three traits are actually hinted at by the very names of the different types of tzara'at:
Se'et
Se'et (שאת) is from the same root as mitnaseh (מתנשא): lase'et (לשאת)
Se'et always means gavoah [high, above, tall] and this kind of tzara'at comes because of haughtiness, which "towers [mitnaseh] over every head"....
Sapachat always means "externally attached" [tefelah] and it comes because of coveting the acquirement of money, which attaches to a person from the outside. And it isn't internalized within a person at all like the rest of the character traits.
And thus writes the Rambam in Shemoneh Prakim L'Rambam, that all the assets--like wisdom, strength, and virtuous traits--all of them become internalized by the person and adheres within him except for wealth, because that is attached to a person from the outside and does not adhere to him at all....
Baheret means "white spots." And this kind comes because of one who speaks lashon hara and whitens the face of his friend [by causing "the blood to drain from his face" in shame]. And this causes white spots....
Why is a kohen the perfect agent to heal tzara'at?
The Kli Yakar explains:
Anyone who is from the seed of Aharon possesses within himself three good traits that are the opposite of those [the above-mentioned]:
1) Because the sin of the tongue causes strife and every plague, and will divide brothers, therefore, Aharon shall come—who held the trait of peace—and shall heal it—because he loved peace and pursued peace.
2) And likewise, the sin of haughtiness is worthy to be rectified through him [Aharon Hakohen] because he was the most humble as is written in Cholin 89a....
3) And likewise, the sin of coveting money was also not within Aharon because the kohanim did not have a portion and an inheritance in the Land, and they did not have more than they merited from the High Table [Shulchan Gavoah, i.e., the Mizbe'ach]. And so they were complete regarding the trait of being satisfied with whatever one has [histapkut] and they possessed no envy because their entire inner goal was to gather and to bring together.
Therefore, it is only fitting that these three types of tzara'at could not be purified by means of anyone but a kohen.

(For a powerful example of Aharon Hakohen as this ideal, please see the second section within Parshat Shemini.)
This is my own translation and any errors are also mine.