Short answer; don't.
Based on what you've said it's not that his character is a problem, just that his theoretical goal is untenable. Make it clear to him exactly what sort of trouble he'd be in for if he decides to have his character pursue that vendetta. Let him know it would be impossible (or almost impossible, depending on your GM style) to fulfill and what sorts of challenges he might face keeping that secret. Then ask him one simple question.
Do you still want to have a vendetta against the Emperor?
If he says yes, great - problem solved. He knows what he's in for and he's decided that that'll be fun. Heck, it will probably be more fun for both of you. Cr0m makes the suggestion below that if you're looking for goals from players that you have him set a more immediate "first step". If he says no, suggest some more campaign-appropriate people to have a vendetta against or help him think of another character concept if he needs 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
Talk to your DM
This is really the only viable solution other than leaving the table (which I'll get to next.) I wouldn't 'gang up' on the DM about this, but broach the subject with regard to wanting to better understand how they (the DM) is generating story and how much impact the PCs have on the world and how much impact the world has on the PCs.
Session Zero
I'd definitely suggest you approach the DM and ask for a Session Zero to discuss the expectations from the DM and the players about how the game and story would run. This is a great way to introduce the issue(s) you and the other player are experiencing.
You can set this up with the DM by just stating that you'd like a better understanding of the world and their game and that there are some questions about it. You can also say that given the game has been going on for about 8 months, it's a good time to make sure everyone's expectations are being met. This includes the players expectations of the DM/game and the DM's expectations of the players.
Sample Questions
But what if there is no progress?
Then you need to ask whether or not you are still having fun at this table. It is still possible to enjoy the table and the game even without your backstory or character growth. But if that isn't the fun you're having, then your only option may be to walk away - and that's okay! Everyone should be having fun at the table. It may not be the fun you expected, but as long as you're still having fun, then let it continue (You just may need to shift your expectations in order to do so.)