"I like no bread."
Like saying "I like no one", this simply means that the speaker doesn't like any kind/ type of bread, and he dislikes bread, in all its variants. If you really think about it, it doesn't have a very different meaning from 'I don't like bread.'
However, "I don't like bread" is a more generalized statement, when compared to the "I like no bread".
But the former can also mean that the speaker isn't fond of bread, but may like a particular type of bread. This is only valid when "I don't like bread" is a part of whole sentence. Consider the following sentence:
"I don't like bread, but I like the ones you make."
This means that the speaker has a general dislike for bread, but likes a particular bread, made by the listener. But if he had said "I like no bread", then it clearly means that he likes none of the variants of bread, irrespective of type, shape or flavor.