Please note that this list is the best and worst interpretations of the source material, as the Joker is and was an established character in the Batman mythos.
First, here's a brief history of the character: in 1940 the Joker was born in Batman #1, and is actually based on Conrad Veidt's character in the 1928 film "The Man Who Laughs" (a film based on the novel of the same name by Victor Hugo), in combination with quintessential illustrations of the Joker playing card. You can see the resemblance below, with the comic issue bearing the same name as the Hugo novel.
With this in mind, we will examine approaches and styles from voice acting to live-action, and see who reached the closest to what we, as the readers, would imagine the Joker would act and sound like on-screen. This is in no way based on the actors themselves, only their interpretation of the original character.
Mark Hamill, Various (1992-2016)
It's no secret that Mark Hamill is Luke Skywalker, but to many fans of the Batman comics he is more the Joker than Luke. From "Batman: The Animated Series" to "The Killing Joke," Mark has played the Joker for nearly 25 years. His interpretation is rumored to be based on the mannerisms of Anthony Hopkins as Hannibal Lecter, and Jerry Lewis. Combined with the comic chaos of the source material, you truly have a force to be reckoned with. Hamill captures the horror of a man who is so brutally aware of his own insanity and lives on the edge of the reality. His legacy is so archetypical of the character that his voice is how you would expect the Joker to sound. Perhaps he gets the least criticism, as he is not donning the costume, but I recommend watching his performances of the Joker on-set so you can see the mannerisms that the animation is based on.
Jack Nicholson, Batman (1989)
Jack Nicholson's Joker was the first Joker we saw on screen since the Adam West days, and the look and style of the performance lives on to this day. Very similar in wardrobe to the comics, Jack's Joker follows the story of Jack Napier (the Joker) and takes us through his days in the mob to the classic bath of acid. Instead of being the guise of the Red Hood, Batman is well aware of his identity and attempts to save him from the fall. This sets up a struggle between Jack and Michael Keaton that will long be a well-accepted on-screen depiction of the Joker and Batman. Jack, in his role as the Joker, captures the cunning sophistication of a serial killer whose sole purpose is to drag as many people into his storm as possible.
Heath Ledger, The Dark Knight (2008)
Heath Ledger's performance in The Dark Knight is truly something to behold. A method actor, he brought many things to the role. Although this is my favorite portrayal, Heath's approach to the character brought something that was completely outside of what we would expect to be the Joker. It's almost as if he was playing a completely different character. This film replaced the acid bath with a bad edition of a makeup tutorial, which (in my opinion) is an element of the Joker's origin that cannot be taken away. I believe Heath offered chaos and unpredictability to the role, but the elements of the cunning gangster and mannerisms were missing from the approach. Nonetheless, Ledger brought everything he could to this role and the performance itself was truly brilliant.
Jared Leto, Suicide Squad (2016)
Jared Leto's recent performance of the Joker is probably even more out of the box than Ledger's Joker. Instead of the comic gangster, we receive an insane thug with a clown getup. I do not think this was necessarily Leto's doing, but if they were seeking a different approach, they may have went in the wrong direction. I would like to see an origin story to this new Joker to understand the tattoos, metal crowns and such and why they took this kind of approach. In short, I see this Joker as less of an intellectual force to be reckoned with and more of an unpredictable force of brute power. Leto has reportedly expressed anger that the film kept out a majority of scenes they taped with this character, which, in "Suicide Squad," is not anything more than a foil glossed over the main plot. Hopefully we can see more of this maniacal Joker, to shed light on the reasons behind these character choices.