The rule is almost certainly based on ECL, just as how Challenge Rating is based on ECL. Even if not explicitly stated, it's apples-to-apples common sense to assume such. There's no reason a level 14-powered Drow needs to get treated as a level 13 character.
The Drow's player should also consider this, if they really feel slighted: http://www.d20srd.org/srd/variant/races/reducingLevelAdjustments.htm
It's not an official rule, but your GM would probably be willing to run with it.
I've played in and run evil campaigns of various sorts in both 3.5 and 4e (though not 5e, I think my learning will transfer), and run into a lot of problems: My Guy Syndrome comes up a lot, as does a tendency to default to a regular D&D storyline only with more stealing of spoons and kicking of puppies to remind ourselves we're evil. Sometimes an evil campaign instead descends into over-the-top motiveless violence until there's no story at all. There's a whole host of at-the-table and in-the-story issues, and I tried many different strategies to address them. Eventually I came up with a framing device which works well for us in avoiding these problems:
Provide the PCs with a Master to guide them toward orchestrated works of Evil.
Start the game with the PCs as underlings/minions/hirelings/apprentices/etc of a powerful evil NPC. The Master has a complicated Evil Plan and he tasks his minions to enact various parts as the Plan progresses: "Bring me the soul of a hound archon," "Raze the border keep," "Steal the Apocalypse Gem," "Help a spy infiltrate the paladin's ranks," and so forth, tailored to the PCs' abilities.
This provides the party a reason to work together despite having different agendas (and working together will hopefully bond them as friends so that they want to continue as a group) and establishes small achievable evil goals that accumulate into an Epic Evil Event.
All you need to do is ask the players to make sure their characters have a good reason to work for the Master: The serial killer likes having his rampages subsidised (and the Master protects him from the Law); the necromancer seeks to learn from the Master's experience and gain access to his libraries of forbidden lore; the mercenary's in it for the money and benefits.
Eventually the Apprentices will surpass their Master.
Expect the party to betray their Master at some point, hijacking his Evil Plot for their own gain: this is not only expected, but awesome. It's the Master's Evil Plot, not yours, and the story isn't about the Master--it's about his apprentices. Consider the Master to be training wheels for evil, setting an example which the party can then follow to surpass and overthrow their instructor as they level up.
This works because Evil Needs Goals.
As Ed describes so well and AgentPaper elaborates in the D&D context, evil needs concrete reasons motivating its actions. The Master provides goals and motives while the players find their feet in the new paradigm, channeling and guiding their exploration of what it means to be evil in ways compatible with the D&D paradigm without simply kicking puppies during a dungeoncrawl.
A word of warning: Alignment is tricky.
D&D has a history of the details and nature of alignment sparking major heartfelt arguments, because D&D alignments are not easily (or appropriately) matched to real-world philosophies and moralities; they're narrative simplifications to support the game's conceits and draw their power from storytelling conventions rather than from genuine moral complexity. Exactly what this means and how to deal with it are beyond the scope of this answer (and possibly this site, although there's a LOT of questions on the topic you can look at), but you should be aware it exists and be ready to talk with your players about what "Evil campaign" means to them so there aren't nasty surprises mid-game.
Best Answer
Not really, but that doesn't leave them entirely in the dark
The concept of experience points is an abstraction. As said in the 5e basic rules and on dndbeyond (emphasis mine):
As noted, experience points are the representation of the knowledge and skills the characters develop over the course of their lifetime. A PC in-world doesn't know about experience points, and they don't all of a sudden gain new abilities when they wake up one morning after killing a bunch of stuff. In world, the characters have been training, practicing, or otherwise gaining strength/skill as they go adventuring. What we call a level-up is the point where the character is confident that they can pull off their new skills consistently and accurately.
That being said, characters still have a sense of their skill level
Just like you or me, a character in the game can have sense of where their skills are relative to where they want them to be. A character close to a level up is basically on the verge of a breakthrough; they are almost confident enough to use their new capabilities, but they just need a little more progress. With that in mind, It's entirely plausible that a character would want to take a less dangerous side mission to practice their new skills before needing them in a dire situation, but at that point it's a roleplaying opportunity for how the player perceives their character would act.
Long story short, characters have an approximate idea of how skilled they are, but they don't have a concept of XP or level ups. A character in game would say something like "I'm so close to pulling it off, I just need a little more practice!" not "I'm only 200 xp away from my next level!"
What if I don't use XP in my game?
The Dungeon Master's Guide page 261 offers variant rules that don't keep track of experience points. If you are using one of these rules (or any homebrew rule that avoids XP), nothing about the interpretation I have provided needs to change except for the scale by which we determine how much practice/experience our characters in-world have gathered.
For session-based advancement, the more play sessions a character sees, the more experience they gain in world. That is, we assume that they're practicing and and getting more skillful each session and after some number of sessions, we say out of game, "The character leveled up," and the character in-world says, "Yeah, I can do this reliably now."
Similarly, story based advancement assumes that the characters just keep on practicing all the time, and we say that they have enough practice to pull off their new talent after some story beat chosen by the DM.