In earlier editions of D&D, multiclassing between spellcasters was generally pretty terrible. Spell levels increased in power super-linearly, so losing access to high-level spells was much worse than gaining double the number of low-level spells.
5e made this substantially better - you add together your levels to determine the spell slots you have, so a Wizard10/Cleric10 still gets 9th level spell slots just like a Wizard20; the drawback is that neither class gives you spells known above what each class at level 10 caster can know (5th level spells) - a lot of spells scale up in power if you use them in higher-level slots, so that 9th-level slot is still useful for a big attack, but it's not the equal of an actual 9th-level spell.
However, 5e also introduced a totally different spellcasting mechanic - Pact Magic - and then utterly failed to address multiclassing with it. A Warlock10/Wizard10 just... has 5th level slots. Two more than a Wiz10 would normally have, and those extra two refresh on a short rest, but still, this sucks.
Related to this, the Spellcasting multiclass rules also cover "half-casters" (like the Paladin or Ranger) and "third-casters" (like the Eldritch Knight or Arcane Trickster) - they add 1/2 or 1/3 their levels to a full-casting class's levels to figure out spell slots. But again, Pact Magic has no obvious way to do "half-casters", which severely limits how homebrew can approach Warlock-ish stuff.
But Here's The Thing
The special thing about Pact Magic is that your spell slots regen on short rest, so you don't need too many of them. But you know who else kinda has spellcasting that regens on short rest? MONKS.
When you go look at monk "spellcasting", they burn ki points to do it, which regen on short rest. They learn up to 5th level spells, spread over twenty levels. They can spend extra ki to power up the spell, at the same time as they unlock higher-level spells. They're basically just spell-point Warlocks, all in all.
(The Elemental monk charges spell level + 1 in ki points, but that's pretty widely recognized as crappy. The Shadow monk charges straight spell level. Other monk subclasses with spell-casting stuff also either charge spell level, or do spell level +1 but get extra benefits, like the Sun Soul which can Burning Hands as a bonus action.)
If we were to convert the Warlock over to Ki points, at the spell level = ki cost rate, the Warlock would even roughly keep up with the Monk's ki pool total, maxing out at 20 (four 5th-level slots). The Warlock just gets additional power above 5th-level spells in the form of their Mystic Arcanum, single-use higher-level spells that recharge on long rest. We'll handle those in a bit.
Overall, the Warlock would retain roughly the same power as they have today - slightly higher versatility, as they could cast more low-level spells in an encounter, but often slightly less overall power. (RAW Warlock gets 3 5th-level slots at level 11, equivalent to 15 ki, while this Ki-lock would only have 11, gradually raising to 15 at 15th level. Similarly, RAW-lock gets a fourth slot at 17, while Ki-lock only has 17 points, finally matching at level 20.) The big benefit is that the Warlock is no longer virtually restricted to scaling spells - instead, they can take non-scaling spells and actually get reasonable use out of them, since they'll just always be cast at their normal (low) cost, while RAW-lock has to "waste" the additional power of their higher-level slots.
So How's This Actually Work?
Here's the plain details of ki-using:
Warlocks get a ki pool equal to their level, just like Monks. It refills on short rest. They can cast a spell that they know by spending ki points equal to its level (and can spend additional points to cast it at a higher level).
At 1st level they can only spend 1 point on a given spell. This increases to 2 at 3rd, 3 at 5th, 4 at 7th, and 5 at 9th. This also determines what level of spells they're allowed to learn, in the same fashion as other full casters.
At 11th level they get their first Overcharge: usable 1/long rest, this lets them cast a spell for free, as if they had spent 6 ki points on it. At 13th level they gain an additional overcharge, worth a free 7-point cast; at 15th, another overcharge worth 8; and at 17th, a final overcharge worth 9. (So, by the end they have four Overcharges, each usable 1/long rest: a 6-point, 7-point, 8-point, and 9-point.) Alternately, instead of getting a free cast, they can spend an overcharge to refill their ki pool by 2 fewer points (the 6-point overcharge can be spent to refill 4 points of ki, 7-point overcharge can refill 5 points of ki, etc).
As a class feature, warlocks still learn one 6th-level spell at 11th level, 7th-level spell at 13th level, etc. These spells cannot be swapped out like their other spells known, which continue to be limited to a max of 5th level.
Monks are half-ki-users; they add 1/2 level to the full levels of Warlock to determine their ki limits and overcharges, but still add their full level to determine their ki pool. The full ki-user multiclass spellcaster table is:
|11||5 ki/spell, 6ki overcharge|
|12||5 ki/spell, 6ki overcharge|
|13||5 ki/spell, 6ki + 7ki overcharges|
|14||5 ki/spell, 6ki + 7ki overcharges|
|15||5 ki/spell, 6ki + 7ki + 8ki overcharges|
|16||5 ki/spell, 6ki + 7ki + 8ki overcharges|
|17||5 ki/spell, 6ki + 7ki + 8ki + 9ki overcharges|
|18||5 ki/spell, 6ki + 7ki + 8ki + 9ki overcharges|
|19||5 ki/spell, 6ki + 7ki + 8ki + 9ki overcharges|
|20||5 ki/spell, 6ki + 7ki + 8ki + 9ki overcharges
"Casting" Monk subclasses, like Way of the Elements, can use overcharges earned from multiclassing in a full-ki-user like normal; they can cast their known spells at a higher level, or recharge their ki pool. They do not learn any higher-level spells, however. Non-casting subclasses, like Way of the Open Hand, have no scaling-ki abilities, and so can only use overcharges to recharge their ki pool.
Interactions with Normal Spellcasters
First, multiclassing a ki-user and a spellcaster partially counts for both; your ki-user levels count ⅓ for the spellcasting multiclass table (or half that for Monks and other half-ki users), and your spellcasting levels count ⅓ for the ki-user multiclass table (or half or third that for lesser casters) and for the ki pool.
Second, ki points and spell slots can be spent fairly interchangeably. If you know a spell from a spellcasting class, you can cast it by spending ki equal to the level of slot you would otherwise use (subject to your normal ki spending limits) or by spending an appropriate overcharge to cast a spell at 6th-level or higher; similarly, if you know a spell from a ki-using class, you can expend a spell slot of the appropriate level to cast it instead.
If a class ability would let you use a spell slot for any non-casting purpose (such as Paladin's Smite, or Sorcerer's metamagic pool recharging), you can spend ki equal to the desired slot's level (again, subject to your ki spending limits, or spending an appropriate overcharge for higher-level slots); similarly, if you have an ability that costs ki, you can instead expend a spell slot of a level equal to or higher than the ki cost.
Interactions That I Think Are Fine
Ki-locks mostly function like normal warlocks, but their interactions with two other spellcasting classes do change a little.
The Paladin/Warlock combo relies on quickly-recharging warlock slots to power more frequent Smites. The only change in using Ki-lock is that the Paladin can do more lower-level smites; a Pal3/War17, for example, would have 17 ki points, potentially powering 17 +2d8 smites per short rest, versus the RAW-lock which gets 4 +5d8 smites per short rest. The Ki-lock can also burn all their overcharges to recharge an extra 22 ki points per long rest, for more smites, while the RAW-lock is limited to using their Mystic Arcanum for their original spellcasting purpose.
So, theoretically this just means that a Paladin could be adding +2d8 to nearly every attack over a short rest. That's useful, sure, but it means they're not opening combat with a powerful +5d8 smite and likely taking an enemy out right away. The raw numbers look bigger, but you really have to take the action economy into account when evaluating this sort of thing. The weaker, more frequent smites probably roughly balance out with the smaller number of more powerful smites that the RAW-lock is restricted to.
(That said, the Ki-lock still can open combat with a big smite, then use small smites later in combat, which is probably a best-of-both-worlds thing. Impact unclear; it's probably still usually better from an action-economy perspective to do larger smites less frequently.)
The other interaction is with Sorcerer; the "Coffee-lock" can unweave their Warlock slots into metamagic points repeatedly over multiple short rest, and re-weave them into Sorcerer slots that last until a long rest. This interaction is mostly just a degenerate rules-abuse that isn't worth explicitly disallowing in rules, in favor of just house-banning such nonsense, but Ki-lock doesn't actually make it any more powerful. A 10/10 mix can produce 13 metamagic points out of ki every short rest, producing a 5th level slot and a 4th level slot; a RAW-lock can only produce 10 (for a 5th and 2nd slot), but ➀ a RAW-lock can produce 15 points per short rest at 11th level; they're just right at the cusp of a big power-gain, and ➁ Pact Magic/Spellcasting multiclassing is absolute shit in the RAW rules; if you use the "each counts ⅓ to the other" multiclassing rules I list up above with RAW-lock, you immediately get the 15 points per short rest. (And I recommend doing so; the ⅓ rule actually works really well overall.)
So overall, the multiclass interactions seem to be well-handled and nice.