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.
Give them 'phantom levels'
I ran a game a few years ago, where we had a brand new player (to our table and RPGs in general) join our level 7 party. This was a similar situation to yours, as I had to get them tough enough not to get perma-dead by the first enemy they encountered, but they would have been overwhelmed if I had them make a 7th level character.
What I wish I had done in that situation, and what user T.L.D. has done (for previous editions) was give them the HP, Armor, and Weapons of a level 7 character, with the features of a level 1 character.
My suggestion to you is that they start at level 1, but have the HP that their character would have if they were the same level as your lowest leveled player at the table (or the level of the table if that's how you're doing things).
Then give them any mechanical bonuses commensurate to the party's level that do not increase the complexity of their character. This may include:
increased proficiency bonus
increased HP
better armor (magical if appropriate, limited to +1's and +2's)
better weapons (magical if appropriate, limited to +1's and +2's)
class features that do not induce choice or increase complexity (e.g. Monk's level 6 feature to overcome resistance to non-magical attacks)
Then, after each session (or two), they replace 1 phantom level with a real one. They do not get more health (you already gave that to them), but they add their class features, feats, spells, etc... that a character of their level would have.
This way, you don't have to be afraid to hit them, but they can still learn the character at a normal pace. Obviously, this works best for more martial classes vs casters.
Best Answer
You should scale the encounters to increase the difficulty for your party.
The ability to scale encounters to your party is very important when you do not intend on following a campaign to the T. The DMG actually clarifies how to do this for your party on page 82. Where it tells you how to gauge your party's difficulty rating by XP values. Within this section of the book, you can adjust the encounters appropriately.
The way to do this by the book is to gauge the combined XP difficulty of your party:
If you have seven players who are level 7, the XP Thresholds By Character Level table specifies:
A Hard encounter for that party would have a threshold of 1100XP * 7 players = 7,700 XP.
A Deadly encounter, by this table, would have over 1700XP * 7 players = 11,900 XP.
When creating the encounter, gauge how many creatures it has, and use the XP from their statblocks in the monster manual (or elsewhere) to determine if they meet this threshold. In this case, let's try using ochre jellies as a test creature for this encounter (though a harder encounter would have more variable creatures than just one type).
The Ochre Jelly has CR2 (450XP). The table on p. 82 of the DMG specifies that you do the following to gauge encounters:
So normally 7 Ochre Jellies would only be 3150 XP. But with the creature quantity multiplier, 7-10 monsters is 2.5X. Bringing that XP total to 7875. This would constitute a Hard encounter by the reasoning within the DMG.
\begin{array}{ll} \text{Number of Monsters} & \text{Multiplier} \\ \hline 1 & \times 1\\ 2 & \times 1.5\\ 3\text{–}6 & \times 2\\ 7\text{–}10 & \times 2.5\\ 11\text{–}14 & \times 3\\ 15+ & \times 4\\ \end{array}
Ultimately, it is your job as a DM to scale up the encounters if the party has improved beyond what the campaign has originally stated. This is a tool provided by WoTC to help DMs determine how to cater to custom campaigns and/or premade campaigns that are adjusted.
I personally find that Deadly level encounters are the only way to give my players a run for their money. Anything below it can be made trivial pretty easily.