Ways to solve the problem of a silent character
I've always wanted to make a non-verbal character...I want to make an in-world way to make communication possible.
This need not be done via a familiar, although a familiar can be a way to do this.
Communicate via Telepathy
For a limited form of communication, the Awakened Mind feature of the Great Old One(GOO) Warlock can work to solve your problem in a different way: you don't need a familiar, and you can be generally "silent," communicating with one party member at a time using that feature.
... You can telepathically speak to any creature you can see within 30 feet of you. You don’t need to share a language with the creature for it to understand your telepathic utterances, but the creature must be able to understand at least one language.
A problem that you will run into as a Warlock: spell casting is a key part of the character and many spells require verbal components. You can mitigate this in a variety of ways and I'll offer four approaches to your general problem that needs solving. (I will guess that you don't want to be a bard ... ). With these as examples, I am sure that you can come up with others.
1. Warlock 1 (GOO) / Sorcerer X
When you get to level 3 Sorcerer, make sure to take the Subtle Spell meta magic so that you never have to use the verbal spell components as long as you have meta magic points. This may be criticized as sub optimized, since it costs a further resource to cast a spell, but it fits your theme. The trick is going to be getting to character level 4 without using spells that have verbal components. Even basic cantrips like ray of frost and fire bolt require verbal components. How far do you want to take this before you get that 3d level of Sorcerer?
2. Warlock 1 (GOO) / Paladin X The Silent Knight
The Strong and silent type. Start as a Warlock (GOO) and communicate sparingly using Awakened Mind. Then begin your progression as the Silent Knight who speaks mostly with his actions. I suggest the Paladin since Charisma is a key paladin ability score for both classes, but you could do something similar with any Fighter Sub Class. With this Paladin+Warlock mix, you may want to add one or two levels of Warlock after Paladin 5 (second attack) for a variety of reasons, but this should work out well enough as a general character concept. You let your sword / warhammer / mace / lance do the talking.
3. Warlock, GOO, Pact of the Chain
As noted in Darth Pseudonym's answer, the familiars available to the Pact of the Chain can speak for you (once you get to third level), and you can use the Voice of the Chain Master to increase the effectiveness of that means of communication.
4. Minor Illusion: I don't talk, but my magic does
For any class that can use cantrips, or by using the magic initiate feat, choose the cantrip minor illusion, and only communicate using that spell. This cantrip has no verbal component.
You create a sound or an image of an object within range that lasts for the Duration. The Illusion also ends if you dismiss it as an action or cast this spell again.
If you create a sound, its volume can range from a whisper to a scream. It can be your voice, someone else’s voice, a lion’s roar, a beating of drums, or any other sound you choose. The sound continues unabated throughout the Duration, or you can make discrete sounds at different times before the spell ends.
For example, you could make a Variant Human monk with minor illusion taken by the Magic Initiate feat.
For another example, you can create a forest gnome PC, monk, who has taken a vow of silence. That sub race gets minor illusion as a class feature.