Moving on...
"The Bad Guys"
CHAOTIC EVIL (CE): "The Anarchist" and/or "The Mindless Savage" A Chaotic Evil character has, very simply, a one track mind- "Bring about the downfall of anything you can." Kill it. Smash it. Destroy it. Corrupt it. Enslave it. So long as you are stronger, smarter, more powerful than it is, then you can do whatever you want.
Ethics mean nothing. Morals mean nothing. Hurting others to get what you want means nothing. The CE character is out for nothing but making themselves "better" (read as "more powerful") than others and cares for no one but themselves.
Peace is a dream for the weak...and the weak, by definition, deserve to be enslaved or destroyed. The only truth is the strong survive. The only things that matter are those that make you stronger/more powerful...Those people, places and things that don't, or get in your way, "Kill it. Smash it. Destroy it. Corrupt it. Enslave it." Society/the world is brutal and you have to be too if you want to get anywhere. Cruelty is a trait to be respected if not admired.
A character would, I think, not so much choose to be this alignment as be indoctrinated into it...through cruel circumstances/upbringing/brainwashing or is simply, psychotic, deranged, indescribably selfish and, well, evil. CE enjoys the savagery of the world and promoting/contributing to it.
NEUTRAL EVIL: "The Villain" and/or "The REAL Bad Guy" The Neutral Evil character has the ethical ambiguity of the Neutral Good. Lying, cheating, telling the truth, playing by the rules...order, freedom, law and chaos have very little interest or concern for him. So long as he gets what he wants, his ends are met.
The morals of NE are the quintessential villain. He is out for his own gains, the means to those ends are of no consequence. This is not the "promote evil because evil is all there is" of the CE nor the "organized evil to increase one's power" of the LE. This is pure and simple selfishness without concern or compunction for any other person, cause or society. If it does not concern (or preferably further) MY neck, MY pockets, MY whims, MY desires, then it does not matter to me whether I slay that golden haired child for stepping in your path or overthrow the tyrant in the duchy next door.
Cruelty, oppression, destruction, all of these things can be entertaining or even "fun"...or not, but they happen and if necessary to his goals, so be it.
A character who wants to be NE is, just that, selfish. Maybe cruel. Maybe dethhhhhspicible. Maybe psychotic. Maybe a craven coward looking to achieve his goals through service to some greater power/figure/organization. But selfish to the point where the concerns of others are not, nor ever outweigh, his own. I tend to think most assassins and evilly-aligned thieves are/would be NE.
LAWFUL EVIL: "The Megalomaniac" and/or "The Enforcer." Power is the absolute. Power is the key to getting what you want. Exercising that power and increasing it is the most noble endeavor. The enforcement of your power in the cause of Order and Law is completely justified.
There is an "honor" about the LE. They are usually willing to "make a deal." She will keep her word...to the letter. Follow the chain of command...unless that chain shows a weakness that can be exploited or needs to be removed. Maintaining Order is the key to keeping and/or increasing her power. One's ethics matter.
One's morals, however, do not. Another creature's worth is solely based on their power and what their power can do for the character. Weakness is repugnant. Betrayal possibly worse. Punishment is swift, potentially cruel and most likely final. "Mercy" is a whine of the weak and the weak, by definition, need to be structured and controlled to serve the greater glory and power that is the ultimate cause of the LE character.
A character would be LE because they wish to "rule the world!" or they simply seek to gain power for their own glory and/or that of whatever master or organization they serve. Lawful Evil characters are akin to the Lawful Good, in my view: Anti-Paladins (which I just usually call "Paladins...who happen to be Lawful Evil"), Clerics of Lawful Evil gods (and/or Devils), and Monks (who have most likely gone rogue from their temple to seek out their own personal power but maintain the discipline - the order of "law" -they've been taught to keep up their fantastic martial prowess)...you can throw in there any rulers, politicians and nobles of any-sized realm looking to increase their personal influence and power.
EXAMPLES:
CE: Sauron, The Joker, Darth Vader, Spike from Buffy the Vampire Slayer (originally)
NE: Saruman, Morgan Le Fey (as told in the original text), The Penguin, Catwoman
LE: Magneto, Lex Luthor, Victor from the Underworld movies, Victor Von Doom for that matter, the Imperial Emperor.