Yes, this spell will control a lich.
Control Undead states:
This spell enables you to control undead creatures for a short period of time. You command them by voice and they understand you, no matter what language you speak. Even if vocal communication is impossible, the controlled undead do not attack you. At the end of the spell, the subjects revert to their normal behavior.
Intelligent undead creatures remember that you controlled them, and they may seek revenge after the spell's effects end.
So "Intelligent undead" can be controlled; however the Lich has several lines of defence against this spell.
- Max 2 HD/Level spell limit. Although admittedly not much of a barrier.
- Will save. The lich is going to have a VERY large bonus for this.
- Spell resistance. And probably this as well.
- Spell Immunity, Greater. Any self-respecting lich will know this spell and have "control undead" as one of the spells it's immune to. Wouldn't you?
Note also, any lich is going to know this spell and have ways around it. The only thing the lich is restricted from (initially) is attacking the caster this isn't going to stop them (until commanded) from...
- Teleporting away.
- Casting silence around themselves so they can't hear you
- Attacking any of your allies.
- Perverting the words of anything you say. "Where is your phylactery?" (The lich answers in Abysmal, which the player doesn't speak.) "Tell me in Common where your phylactery is?" (The lich says "It is under the sacred cloth of my father.") and so on... all buying it time to get away.
- And so on.
It's not a hands-down game winner against a Lich,
A warlock following a patron of The Undying
In the new Sword Coast Adventurers Guide (page 139), The Undying patron is detailed. This seems to suit your needs exactly, although you might not be able to change from one class to another without your DM's permission.
Listed as example Undying are actual liches like the lich-queen Vol, Vlaakith the lich-queen of the githyanki, and "sentient wizard zombie" the deathless wizard Fistandantalus, among others.
Your alignment is not tied to your patron, and you can read about that here.
The Undying expanded spell list includes spells like contagion, death ward, false life, ray of sickness, speak with dead, and other lichy type things. The following features are paraphrased, so you'll need to grab a Sword Coast Adventurers guide to get specifics.
Among the Dead
At 1st level, you learn spare the dying and have advantage on saving throws against disease. Also, when an undead creature targets you, they make a Wisdom throw against your spell DC and can't target you on a failed save. They are immune to this for 24 hours if they succeed or if you target it with an attack or spell.
Defy Death
At 6th level, you gain 1d8 + Con hit points when you succeed on a death saving throw or use spare the dying once per long rest.
Undying Nature
At 10th level, you can hold your breath forever and don't need food, water, or sleep. You also age 10 times slower and can't be magically aged.
Indestructible Life
At 14th level, you can use a bonus action to regain 1d8 + warlock level hit points. You can also reattach severed body parts when you use this feature. can use once per short or long rest.
So, while not technically immortal, this is pretty close for RAW and requires no homebrew unless you change from your current class to a new one. An elf Undying Warlock can live to be about 7000 years old, which is really enough time for you to roleplay until you actually die of old age in real life.
Best Answer
In general, they have few specific weaknesses beyond those all undead share. Specific liches can of course have unusual weaknesses.
Since this is for 2e, I recommend looking at Van Richten's Guide to the Lich. It is designed specifically for Ravenloft, but it is generally compatible with almost any of the 2e settings. It goes into background on liches and lays out a number of variations which a DM can use for making custom liches, including unique weaknesses that a particular lich may have.
Of particular note, it suggests that many liches may need to periodically perform elaborate rituals to continue to maintain themselves. Those rituals can be exploited by patient lich hunters because the requirements of the ritual often force the lich to venture forth from their lairs to either acquire substances needed for the ritual itself or to be in the right place at the right time to perform the ritual. In short, they provide an opportunity for adventurers to ambush the lich at a time when the lich is away from its base of power.