Claude is mute because of a reason. Have any of you played through GTA2 ever? You play as a guy in top down view, doing jobs for only 3 possible gangs within the city, and you're not doing out of the gangs interest, you're doing it out of money, but then you also can do other gangs jobs, even if it's the opposing gang. GTA2 and GTA3 have similarities between each other. In GTA2 you play supposedly as Claude Speed (Although when you watch that small movie at the beginning, you see he's Asian). In GTA3 you play as someone else, but is called Claude Speed. Both GTA's have gangs you work with, in which you have to play as a Neutral Character, because if you worked with other gangs and you followed interest with them, you wouldn't be called neutral would you? Being neutral means you only care within the mutual interest of the money, not the reason for the gang.
So the reason he's mute, is because if he wasn't, he wouldn't be neutral, plus he probably would alliance with a gang, rather than work as an independent contractor doing jobs for others. And if he ever did talk, he would probably end up getting himself killed. He's better off keeping quiet for a good reason, to not get himself killed. And because he's just awesome being quite, at least you could call him the "Silent Killer".