I have a character who speaks in a sort of telepathic manner through feelings and images, and I imagine you could do a very similar thing with your mute character. It can go a few ways. One is to describe very subtly a few of the images (or gestures) he's portraying, as well as the receiving character (preferably the speaker) translating what it means in their own words. This can be difficult but is especially useful if they're just now meeting and figuring one another out. Another way (can be used in conjunction or alone) is to directly translate in a slightly 4th-wall-break manner (ie, with [ ] brackets) I'll provide a small example of each below:
She turned to look at him. He still hadn't spoken, but was now beginning to move his hands. What was he doing? It seemed like the gestures were very deliberate. Oh! Sign Language! He seemed to realize her confusion, and slowed down a little. Let's see… she knew that the salute-type sign meant hello… pointing to himself…some other ones that went too quick to understand…and then he was spelling! A…L…A…N…! That must be his name! She quickly followed to repeat "hello" and spell her own name.
Or alternatively:
She knew a few signs, and hoped she could infer his meaning. [Hello. My…..is ALAN] The missing gesture must be name - his name was Alan! She quickly followed with her own introduction.
As you can see, the second one is a lot shorter of a paragraph, but you can be hard to figure out how to embed and you lose some potential insight into the character's head. Sorry I got a bit caught up in the first one and neither is totally polished, but I hope this helps you out!