That is actually not true. In 5e a paladin can choose to swear their oath to an ideal rather than be subject to the approval of a deity but they still have the option to be tied to a deity. Divine smite is .....well it isn't a spell, its divine, which connotes divinity, which is of the gods. You could easily re flavor the oaths to fit drow society. What is good and lawful to a drow or even Lolth would be completely different from what is good and lawful to a human or say, Torm, you get the picture.
Literally any character of any race or class can choose to be tied to a deity. Nothing in the character creation of a paladin has you choose to have allegiance to a god or even gives you that option. If that's the flavor you're going for sure that's fine, but you could do the same thing with a wizard or a barbarian, it's not a part of being a
paladin. You do have 'divine' abilities but in the same way that wizard's and sorcerers are not tied to pacts like warlocks even though they share arcane magic your divine magic need not be tied to a god. Rangers and Druids are an even better example. Druids will not even wear metal armor and disdain society choosing to worship and commune with nature in order to get their nature magic. Rangers still have nature magic and many of the same spells but none of the tenets of that nature worship, their desire for the hunt and familiarity with the wilds gives them their magic. So to does a Paladin's fervor for an ideal above all else give him/her divine abilities.
Honestly though that is my interpretation. There is quite a bit of mention in their class of deity's and even clerics of the same faith, prayer and the like. So by all means if you want your paladins to be more martial servants of the gods then have at it. My main reason for bringing it up is to point out that it is not a requirement by the book as it used to be.
I do really like your idea of portraying the other side of the coin of morality. It's something that many stories, games, books etc. really miss. The fact that even for the bad guys they are doing what they think is good and right. I cringe so much when stories demonize the bad guys to the point where they are knowingly harming themselves and others for no redeemable reason other than they need to be 'evil'. There are a lot of motivations to do awful things but truly great villains are human too. There are so many great examples of this but one two that are good illustrations;
Lex Luthor. I'm not sure which writer it was but by far the best explanation for lex luthor's pathological hate of superman is that he truly believes that superman is an alien invader that is crippling earth's growth. To him as long as we have superman to bail us out we will never mature as a people and realize our true potential. He isn't a crazy megalomaniac bent on world domination. To him he is trying to save the world.
The Chosen(wheel of time). A wonderful series in many ways but one of the definite highlights is getting to see through the villains eyes from time to time. You get to know them and understand that for most they do not want to serve the Dark One anymore and are tired and apathetic after being reincarnated endlessly for the last millennia. They no longer care about the power or the greed that drove them to his service and many regret ever signing over their soul. They are human, or at least where, and are very relate-able. They know what they are doing is wrong and they are either driven mad by the Dark One or are living in fear.
On a side note I find it interesting that there is a whole section about how a paladin can break his oath and how to handle that but absolutely no mention of how to deal with a non repentant cleric or even one who leaves the faith. Do they lose all cleric levels? Do they have to atone? Their power comes very directly from the gods so it would make more sense that their conduct would be judged, but there is no mention of this :/
-that got really long winded tl;dr you don't need a deity but feel free to have one. If you are going to play a morally grey character don't be a boring psychopath.