This question isn't limited to the silk road btw, I've always been confused as to how the character based languages of Asia translated to latin based languages.
There's a small problem with your question: Chinese languages are not character based. The writing system uses characters, but the language itself cannot be character based, since languages are not based on their writing systems.
Like with all languages, children learn to speak well before they learn to read or write. This is true with Chinese people just as it is with French or Italian people.
The way that the language barrier would be overcome between an Italian in China in this period is the same way that it would be overcome by a European person going to Egypt or an early explorer in the Americas. Intermediaries were used when possible, and when not available (such as in the example of the Americas), people would have to learn the languages to communicate, and rely on more basic means of communication before then.
A very similar question was asked just three days ago, which you should have a look at.
You might also be interested in these similar questions that have been asked in the past:
I think the problem is that you are assuming that one trader would go the whole or a greater part of the length of the route. In fact, the route was made up of many traders, each moving along part of the route. A single trader would only need to know his own language and that of the most contiguous areas