To ask for a PostDoc position in the university X, is better to do it by myself or to ask to my supervisor?
Put yourself in the shoes of the professor: what would they like best in a candidate? Contacting them yourself demonstrates more commitment than having your advisor do it for you. There may be cases where an introduction by a third party is best, but it seems to me it's better to use that only if really needed: guy is a superstar, your introductee knows him very well, or you are just trying to strategically probe whether a candidature would be well-received without actually getting involved yourself.

Also, you can get your advisor to follow up if you don't receive an answer, or just to help you when/if they are contacted as a reference.