"Dudley", as you know, is usually used as a proper noun, not a common noun. Most proper nouns don't take articles, so why insert one?
The simplest analysis is that inserting an adjective converts Dudley into a common noun. In other words, "Dudley" starts representing the class of Dudleys: an angry Dudley, a pacified Dudley, a sullen Dudley, and so on. This device has a long history and is mostly used in writing; it is most commonly used to divide a person into personalities (as above) or into ages (e.g. "a young Rembrandt").
As a common-noun phrase, "screaming Dudley" can take either the definite article "the" or the indefinite article "a". The rules for this are the same as for any other common noun.