Official Stats are in Volo's Guide to Monsters
Volo's Guide to Monsters (released November 2016) includes many additional Illithid variants and related creatures. One of these is the Elder Brain.
The official Elder Brain description can be found on pages 173 and 174 of Volo's Guide and includes lair actions, regional effects, a full stat block, and lore on the nature of the Elder Brain.
But Matt Mercer's Elder Brain is homebrewed
Matt Mercer's Elder Brain appeared well before the publication of Volo's Guide, and thus is distinctly homebrewed. Additionally, Matt himself has stated:
...and at full strength [the Elder Brain] is a CR 20 creature.
This doesn't line up at all with the official publication, which only has a CR of 14. A quick search didn't turn up any exact stats for his Elder Brain, only a meager collection from the third party critical role stats site which has the following information:
- Elder Brain First Appearance: 10 K’Varn Revealed
- Armor Class 18 (at weakest), 20 (before party flees)
- 320 damage taken and “not on death’s door yet”
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
According to Volo's Guide to Monsters, that's very unlikely. Specifically in the part that talks about the conversion of a Ulitharids into an elder brain:
This implies that the brain must be harvested immediately after its death, and that it must be not decrepit. This is somewhat incompatible with lichdom, as the emaciated and dessicated brain of a walking corpse could hardly be described as "not decrepit", and it is hard to create a lich (or variants like the Alhoon) from a brainless body.
Of course, nothing says it is impossible for a living elder brain to become undead.