I would recommend using the Monster Vault maths, found summarized on Blog of Holding; or, Monster Maker is a handy app that helps create monster cards, and will work out the maths if desired.
Set out which roles you wish the monsters to take - from the sounds of things it seems like you'll have a mix of Lurkers and Skirmishers, with Soldiers and Brutes making up the majority of the rest. Adding some of the other roles (artillery and controller) will help build variety.
Essentially you will be making sets of 'monsters' to form into encounters. From memory approximately the same number of same-level enemies makes a relatively balanced encounter, depending on the level of optimization and experience of the players. Mix and match some monster types, replacing 1 monster with 4 minions of the same level, and nudge up or down to taste.
For example, for 5 level 1 PCs you could then have 2 'Tough Monks' (level 1 Soldiers), 2 Ninja (level 1 Lurker), 4 mook monks (level 1 skirmisher minions) and 4 ninja mooks (4 level 1 artillery minions).
The DMG 2 has a section on making monster 'themes' - in essence by putting together sets of additional abilities that are thematically linked - such as the Goblins' ability to shift. For instance, you could add an ability to the Monks that acts in a manner similar to the Monk PCs flurry of blows - maybe on a successful attack they deal a small amount of damage to an additional adjacent enemy, maybe 2 damage for a level 1 monster.
One of the easiest tricks for 4e is to simply 'reskin' a monster - presenting for example, a goblin or a kobold as a ninja is just a question of description. Find a monster which has the relevant ability (such as shifting on a miss, those pesky goblins!) and you're good to go.
Generally a 4e encounter is much better with some environmental factors, so I'd recommend adding in some interesting terrain features or traps to spice things up a little. Broadly speaking, avoid adding in higher level Soldiers or Brutes too much, because they can become hard to hit leading encounters to bog down. Lots of minions is useful in making an encounter feel exciting and allowing the PCs to feel capable heroes - it all comes down to description when using minions - you can make the heroes feel awesome or like they wasted their attack depending on how you invoke their sense of adventure.
The monks could have the ability to spread some splash damage as a flurry of blows, the ninja should be tricky to pin down, maybe with a teleport style ability or a jump. They could climb walls with normal movement for a more HK film feel. The tattoos sound like a great idea, possibly you could borrow from Legend of the 5 Rings and each style of tattoo grants an ability - so the monks with flame or dragon tattoos can breath fire. Monsters that spawn other monsters are fairly tough - the Wraith in 4e does something similar I think. Make sure that whatever comes afterwards can't spawn more monsters - maybe it generates a minion when killed that lacks the respawn ability. Alternatively, you could describe them as crumbling and rising once they hit Bloodied, with some new abilities?
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
Technically, yes.
You can trick someone into turning themselves into a lich. But...there's a little more nuance going on here. Skip to the end if you just want the conclusion.
How to become a Lich in 5E.
As you may have noted from the MM, it's rather vague about how liches are created. See here for the full text. In a short summary, we know this:
We get a little bit more information from the module Curse of Strahd. While it does not put a price tag on any of this, it does state that
Nothing in 5E says that you have to make your own phylactery or brew your own potion...CoS simply states you can't make both at the same time. If we go simply based off of this, it is feasible that you could make a phylactery for someone else and brew up the potion for them to drink...only the knowledge of how to make those things has a precedent for requiring a certain degree of power.
But if we look back at older lore
Prior editions.
The most comprehensive guide we have on Liches comes from AD&D2E in Van Richten's Guide to the Lich. This extends the list of requirements as follows...
This is further supported in 3.5E where it says
We also learn that the spells used include the following: Permanency (8th level), Enchant Item (6th level), Magic Jar (5th level), and Reincarnation (6th level) for the Phylactery. The following must be cast on the potion of Transformation: Wraithform (3rd level), Cone of Cold (5th level), Feign Death (3rd level), Animate Dead (5th level), and Permanency (8th level) Note: Spell levels are in AD&D2E terms...and yes, in 2E, Wizards could cast Reincarnate.
Thus we can conclude that since the lich-to-be must enchant the phylactery on their own they...
3.5E dials that back a bit, ruling that you must
Note that 'half-casters' like Rangers and Paladins have a Caster Level equal to half their class level--thus a half-caster cannot reach caster level 11 unless they multiclass into a full caster _for 11 levels). Your Caster Level only counts towards the specific class you're using to cast that spell. A Cleric3/Wizard5 does not have a Caster Level of 8--they have a Cleric Caster Level of 3 and a Wizard Caster Level of 5. So, by 3.5E, standards, only full-casters of 11th level or higher (in the same class) can become liches.
An additional 5E Constraint
This is quite important. In order to maintain its physical form, a Lich must regularly feed souls into its phylactery. According to the Monster Manual...
Imprisonment is a 9th level spell. If you are not able to cast 9th level spells, specifically Imprisonment, then you are not able to sustain yourself as a Lich.
Now, the MM specifies that you become a MM Lich upon completing the lich transformation--but it doesn't make sense that you spontaneously gain full spell progression as a Wizard if you didn't already have it. Thus, if you tricked someone who couldn't cast 9th level spells into turning themselves into a lich, they would only last so long before they degraded into a demilich.
Conclusion
Ultimately, you can trick someone into becoming a lich. The process requires you to make a phylactery, then drink a horrible-awful-magical potion stuffed with a cocktail of poisons, the blood of a sentient creature, and boosted with spells.
There is nothing in this process that requires disclosure of what this ritual will do to the one who conducts it. The knowledge of how to become a lich is so rare and so obscure that it's entirely possible for someone to have no idea what will happen when they follow your instructions. You can lie to them about what the ritual does and rope them into it.
It would be a simple enough lie to tell them that it will let them bind souls into the box that they can draw on for power...could even justify it saying that you can target Evil Creatures--ending their reign of terror then using their souls to do some good in the world. Or just lie with the truth and tell them that it will 'secure' their soul so that even mortal wounds cannot slay them (just leaving out the bit about 'securing your soul turns you into a remorseless undead nightmare').
However, it also requires the lich-to-be to not figure out what you're asking them to do. You're making them carve arcane sigils into a little box, cast a crapton of spells on the thing, brew up a horrible potion mixed with blood...it's very possible for them to suss out that this little box sounds an awful lot like a Phylactery. And given that most Liches used to be Wizards and are thus hyper-intelligent...they're probably going to figure out what you're making them do.
In your case
If you consider older lore...your specific case won't work. Eldritch Knights are utterly incapable of reaching the magical power level necessary to enchant a phylactery and become a lich. And even if they could, the EK's inability to cast 9th level spells means they couldn't sustain their lichdom.
Might I suggest seeing if you can brew up an Arcane alternative to a Death Knight instead? That seems much more fitting for an EK anyway.