I can't take credit for this, but a friend of mine that I game with put this together for a friend playing a Mind Flayer in our current game:
Illithid
Alien, frightening beings of great mental prowess and forceful personalities.
Ability Score Increase. Your Intelligence score increases by 2, and your Constitution increases by 1.
Age. Illithids fully complete ceremorphosis by the age of 9, and have become fully mature and complete in their education by age 15. Illithids live longer than humans, but they do eventually die naturally after hundreds of years.
Alignment. Illithids normally have a natural tendency towards hierarchy, discipline, and cruelty, and have an alignment of lawful evil. Some individuals find themselves developing more-individualistic personalities than their fellows early after ceremorphosis and have an alignment of neutral evil. A minority lack the characteristic sadism of their brethren while retaining their preoccupation with organization, and have an alignment of lawful neutral.
Any illithid of any other alignment, usually the product of unusual circumstances such as mutation, magical alteration, extensive contact with unshackled non-illithid minds while cut off from the networked communal intelligence of their colonies, or intervention by otherworldly forces, is likely regarded by others of its kind as either a disgusting social deviant and pariah, or, at worst, as pollution to be cleansed. Many such illithid are either mentally wiped blank and re-educated or simply thrown out of their home communities if their natures are too open and offensive to their fellows' sensibilities.
Size. Most illithids are at least as tall as tall humans, standing at least a few inches shy of six feet, but their bodies very slender and emaciated, weighing much less than an average being of their stature. Your size is Medium.
Speed. Your base walking speed is 30 feet.
Darkvision. You have darkvision up to a range of 60 ft.
Magic Resistance. You have advantage on saving throws against psionic and magical spells or effects.
Mind Blast. You may use your action to attack anything in front of you with a mind blast. When you use your mind blast, each creature in a 15 ft. cone must make an Intelligence saving throw equal to 8 + your Intelligence modifier + your proficiency bonus. A creature takes 2d6 Psychic damage on a failed save, and half as much damage on a successful save. This damage increases to 3d6 at 6th level, 4d6 at 11th level, and 5d6 at 16th level. After you use your mind blast, you can't use it again until you complete a short or long rest.
Devour Brain. As you consume the brain of another creature, you briefly interface with its mind. You may use your action to attempt to wrap your tentacles around the head of an unconscious living creature or a living creature that died in the last few minutes, then use your alien jaw to open its skull and eat its brain matter. After doing so, you learn at least one piece of useful information the creature knew, and while you maintain Concentration, as though concentrating on a spell, you gain temporary proficiency with a weapon, skill, or set of tools the creature had proficiency with. If it was not dead, using this ability (obviously) kills the creature.
Aberrant Mind. Even those illithids who have undergone alignment shifts via magical means or mutations struggle to understand the thoughts of less-alien beings. You suffer disadvantage on all Wisdom (Insight) checks, even if you are proficient in them, and on any Charisma ability score tests in which you are not proficient.
Languages. You can "speak," read, and write, Common and Undercommon, and can write Illithid, a language so unintuitive to most humanoid minds that only fellow illithids can decipher it without the use of magic by feeling it with their tentacles.
Alternatively, if you play a Dragonborn but reskin it as a Mind Flayer. Just change the resistance to psychic, and the breath weapon to cone and deal psychic damage.
If you also went sorcerer, you could go Dragon Bloodline and take the following spells, changing their damage type to Psychic for flavor reasons:
Sorcerer
Level - Spells
1st - Burning Hands,* Detect Magic
3rd - Agnazaar’s Scorcher,* Enthrall
5th - Fear, Fireball*
7th - Elemental Bane**, Wall of Fire*
9th - Cone of cold*, Dominate Person
* Damage type changes to psychic
** Damage type affected by spell is always psychic
Sorcerers in my games gain access to a kind of Domain Spell list as bonus spells, which is where I came up with these.