[RPG] Are all Liches innately evil?

So here's my issue: I like my character, I like our party, and I don't want to pull a 180 on my character and make him nice or throw away important motivations for him.

Well, it sounds like your character just may be evil, or at least on the evil side of neutral.

That doesn't mean he has to do evil things, especially if he has a reason not to.

And, if he wants to stay with the party, he probably does. You've already had an in-character conflict where one character "stormed out" and another "left the room disgusted." That ought to be a pretty good clue to your character that, if he wants to hang around with these guys, he'd better start to act nice, even if he's only doing it to keep the other party members cooperative.

Even if your character was a complete psychopath who loved kicking puppies, if he was sufficiently smart he should be able to realize that there are situations where it's better to heroically save the puppies instead. And it doesn't sound like your character is anywhere near that bad.

Basically, you're playing a conflicted character. This can be a lot of fun, if that's the kind of thing you like.

It doesn't matter that the cause of the conflict may be (at least initially) external; even so, it's a source of mental conflict for your character. On one hand, your character worships an evil god. Even if his choice is fundamentally based on pragmatic reasons (power!), a pact with Cthulhu will surely have some influence on him, tempting him to more evil acts and means (not that a hunger for power couldn't do that all by itself). On the other hand, he's also fighting to save the world (even if it might be for his own ends), and has teamed up with a bunch of more noble, good and squeamish types to do so. This means that, whatever means he might want to employ in the pursuit of his goals, he now also has to consider their effect on his fellow party members.

And, of course, once he starts to consider the way his actions are perceived by others, he might also gradually come to realize that there's also a side to himself that doesn't like what he's doing. Maybe not instantly, but after a while. Morality has a funny way of growing on you like that — from "I have to be nice or I'll get punished" to "hey, other people are actually nicer to me if I don't act like an asshole" to "you know, I really should be nice just on principle, because it makes the world a better place."

None of this means "pulling a 180" on your character, or at least, not in a way that he wouldn't have good in-character reasons for. Sure, the conflict with the other party members (and possibly subsequent reflection) might be the trigger that makes your character realize that he needs to drastically change his behavior in order to achieve his goals, but his motivations will still be the same as before.

Paladins in D&D 5e don't have to be of good alignment.

Firstly, even in earlier editions of the game, a tiefling could be of any alignment, even lawful good, and become a paladin. In D&D 5e, it's even easier as there are no longer any alignment limits or racial restrictions on any character class.

In particular, the Oath of Vengeance paladin (PHB p87) may suit that character concept.

The Oath of Vengeance is a solemn commitment to punish those who have committed a grievous sin. When evil forces slaughter helpless villagers, when an entire people turns against the will of the gods, when a thieves' guild grows too violent and powerful, when a dragon rampages through the countryside—at times like these, paladins arise and swear an Oath of Vengeance to set right that which has gone wrong. To those paladins—sometimes called avengers or dark knights—their own purity is not as important as delivering justice.

Another paladin type especially suitable for lawful evil characters is Oath of Conquest (Xanathar's Guide to Everything p.37):

The Oath of Conquest calls to paladins who seek glory in battle and the subjugation of their enemies. It isn't enough for these paladins to establish order. They must crush the forces of chaos. Sometimes called knight tyrants or iron mongers, those who swear this oath gather into grim orders that serve gods or philosophies of war and well-ordered might.

There's also the Oathbreaker paladin (Dungeon Master's Guide p.97) who is a formerly good paladin turned to evil. This may not fit your backstory if your character was always evil. The Unearthed Arcana content also includes the Oath of Treachery, a paladin who similarly broke a previous oath, although the description suggests it's more for chaotic evil characters.