Charisma can mean many things
The PHB defines charisma as: (PHB 178)
Charisma measures your ability to interact effectively with others. It includes such factors as confidence and eloquence, and it can represent a charming or commanding personality.
Note that this definition does not mention any desire to interact with others--only that such interactions are effective when they happen.
Furthermore, charisma covers a wide range of social interactions, such as intimidation: (PHB 179)
When you attempt to influence someone through overt threats, hostile actions, and physical violence, the DM might ask you to make a Charisma (Intimidation) check.
Someone who has proficiency in intimidation has a very different kind of charisma than someone who has proficiency in deception or performance, for example.
What does this mean for your warlock?
First, your warlock can be as socially awkward as they want; it's just that when they do decide to interact with others, it's very effective.
Perhaps your warlock is very scary to others, and is perfectly aware of that fact. This means they are good at intimidating and scaring other people by deliberately playing up their mental instability, but they know that such tactics don't work in all circumstances.
Alternatively, your warlock could be very social with a certain group (other arcanists, or maybe eldritch horrors?), but not with "ordinary" people. After all, the kind of social skills needed to interact with nobles is probably very different from those needed to interact with demons.
Or, it could be that your warlock is truly mentally unstable and socially awkward in all cases, but something about his patronage with the Old One makes him supernaturally compelling to other mortals. They're already getting all sorts of other weird, eldritch powers, after all.