So this is a segment where I pick an actor/role and try to break down the accent as it relates to the role as well as the actor. Why does the actor have this distinct accent? I think it is interesting because these actors do not sound necessarily "typical" but they command attention and draw the listener/audience in despite sounding different.
Christopher Walken is an iconic actor with a distinct accent. He distinguishes himself on the big screen by the way he talks and you don't need to see him to recognize the actor. He plays King Louie in the CGI-live action remake of The Jungle Book (2006) and his characteristic prosody (the patterns of stress and intonation in a language) is undeniable.
Typical American English phonetic repertoire. He can make each and every English sound appropriately and can distinguish between them.
In part, his accent is loosely a Queens, New York based accent
"th"-stopping - /th/ becomes a plosive. "Dere was a..." for "There was a"
omission of "r" in unstressed syllables.
Alveolar consonants /t/, /d/, /n/, and /l/ are articulated with the tongue blade rather than the tip resulting in a more dentalized. Instead of the tongue tip touching the alveolar ridge, it often touchs the teeth themselves.
Dropping the "h" sound in "huge" or "human".
Non-meaningful intra-word variation . Christopher Walken will raise the pitch of his voice in the word-final position. This does not reflect a meaningful change to the utterance (i.e. not asking a question).
Prosody (The interesting part)
As a child, Walken reportedly erased all commas, periods, apostrophes and exclamation points. Sometimes he'd put in commas and exclamation points of his own, next to words he liked them to be next to. I can imagine that this behavior carries on to his script reading where irregular pauses, stresses and pitch variations are his characteristic trademark 
A wide range of non-verbal facial expressions characterized by eyebrow raises, furrowed eyebrows, widened eyes, lip smacks and lip licks. Coupling that with his unconventional stress pattern and Christopher Walken commands attention. You are drawn into every word he says.
Take a look here at his reading of The Three Little Pigs and watch his non-verbal communication
Here are a bunch of great impressions of the man. See what you agree with and let me know if I've missed anything!