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:
- Liches used to be Wizards
- They need a Phylactery
- They have to drink a potion of transformation (made with the blood of a sentient whose soul goes in the Phylactery)
- Generally had to strike a bargain with some horrible being to learn how to do it
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
The being who gives out this knowledge won't hand it out unless you can cast 9th level Wizard Spells.
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
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...
- While they can commission someone else to make the box itself, the lich-to-be must enchant their own Phylactery. This includes all the runic engravings done inside the box.
This is further supported in 3.5E where it says
Each lich must make its own phylactery
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...
- Must be capable of casting 8th level spells
3.5E dials that back a bit, ruling that you must
The character must be able to cast spells and have a caster level of 11th or higher.
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...
A lich must periodically feed souls to its phylactery to sustain the magic preserving its body and consciousness. It does this using the imprisonment spell. Instead of choosing one of the normal options of the spell, the lich uses the spell to magically trap the target’s body and soul inside its phylactery.
A lich that fails or forgets to maintain its body with sacrificed souls begins to physically fall apart, and might eventually become a demilich.
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.
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.