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.

How can I encourage...?

First, though not tagged D&D5e, the answers to these questions contain much valuable advice. I suggest you peruse them:

• How can I encourage role playing in a custom setting?
• How to introduce and encourage role playing in non-roleplayers?
• How do I encourage new players to be more descriptive in their actions and intents?
• As a GM, how do I encourage a party to interact with each other?

However...

There are many different games we play.

^{WARNING: AngryGM liberally salts his excellent advice and analysis with rude and vulgar language.}

It sounds like you want to play the game where "we throw ourselves into someone else's shoes and talk like them."

It doesn't particularly sound like this player wants to play that game.

That's okay.

At any D&D table of \$N\$ players there tend to be at least \$2N+1\$ games being played, in my estimation.^* Not every player equally enjoys each of those games.

The key steps for you to take are to discuss the following with that player, possibly with the entire group participating:

1. I don't see you engaging much in "acting" scenes--do you enjoy that time? (The player may just enjoy watching the show play out.)
2. Do you want to be more active during those scenes?
3. Are there parts of the game you wish we'd be spending more/less time on? (This is a good question to ask all your players, every session.)

And then use the answers to these questions.

Coda: System Matters

I feel I'd be remiss in not adding this: recognize that you're also playing in a system that's not really optimized for that game. Sure, D&D handles it, but it's not built around the sort of roleplay you're describing.

If your player really does want to expand their acting-game, and the things you can try within D&D don't work out, then you may want to consider fantasy-adventure roleplaying games that are built around play-acting.

^{* - note that number snuggles somewhere in between "more games than players" and "not as many games as pair-interactions" for \$N>3\$. \$N=2\$ and \$N=3\$ are special cases.}