Hugh Laurie again, but in his roles in Blackadder.
I wasn’t really thinking of actors who play a character who uses an accent different than their own. Listing them isn’t really an exciting discussion or thinking exercise now, is it? 
I do like the Hugh Laurie and David Tennant answers so I am torn! Never seen Walking Dead but British to Southerner does sound like hard work.
God answer on Bronson Pinchot!! I have been re- watching that show actually. Duh!
I think anyone that has to do extensive makeup for every episode has got it hard so kudos to all the aliens and zombies and Throne people. Danny Jean-Jules and Robert Llewelyn from Red Dwarf!
I totally forgot David Suchet as Pirot. He wears a fat suit, speaks in a unique accent, walks funny and has as many mannerisms as Michael Richards did.
Quark and Worf had to do hours of makeup everyday
I guess not really hard “acting” but it’s something
Julie Kavner has said it’s physically difficult to do Marge Simpson’s voice. I’m guessing she never thought she’d be doing it for twenty-seven years.
Another one is Damian Lewis. I watched him playing an American on Band of Brothers and he did the accent so naturally I never realized he was actually British until I saw him somewhere else.
Nice one!
Dr Marvin Monroe was too hard for Harry Shearer to do so they killed the character
On the Showtime sitcom Episodes, Matt LeBlanc plays himself, sort of, acting in a television series adapted from a critically acclaimed British programme. Here, he (or more precisely, his character) is ranting at the British writers of his show, “I’m so sick of these British fucks coming over here and stealing our parts. How is that even legal? Huh? It should be ‘You act in your country and we’ll act in ours.’ But nooooooo. They sneak in here with perfect American accents like some fucking magic trick. And no one even knows until it’s too late. And suddenly you see them on some talk show and you’re like, ‘Holy shit. That guy’s English?’ They’re in our TV shows. They’re in our movies. There should be Congressional hearings or something. Fucking British people. Stay the fuck home. No offense.”
Estelle Getty as Sophia Petrillo. Wearing make up to make her appear twenty years older, and she had severe stage fright.
Even better is when one clone is disguised as another clone. If you watch closely, you can see for example Sarah all dressed up as Rachel but still walking like Sarah. I was quite proud of myself during one scene when I figured out that Helena was playing Sarah, right before they revealed it was so.
Anthony Star, Ivana Milicevic, Hoon Lee, and Matthew Rauch of Banshee.
Hugh Dancy in Hannibal. Eva Green in Penny Dreadful. Clark Gregg in Agents of Shield. My favorite is Andreas Katsulas as G’Kar.
I’ve only just started this, but was very struck very quickly as to forgetting it was the same actress playing all these widely different roles. I just watch it now assuming they are different actors, confuses me less.
I’m not sure who I’d pick. Maybe someone from one of the action TV shows, where they have to keep in top shape and be able to do stunts every week? Charlie Cox from Daredevil would be a good example: a British person doing a convincing American accent, has to be in very good shape, does some stunts (though has a stunt double of course to do some of the stuff), and plays a blind person who can use other senses to get around but also has to pretend to be a more normal blind person much of the time. He manages to do it all pretty well, and also do the emotional and character driven parts.
Special notice for the episode where they run into Queen Victoria in Scotland, so the Doctor fakes a Scottish accent, but as the episode goes on slowly slips into his normal English accent without realizing it. I thought that was very well done.
Neil Patrick Harris as the womanizing asshole Barney in How I Met Your Mother is an absolute stretch. Not only is Harris an all around nice guy, but he’s openly gay, in a monogamous relationship, and a father to twins.
Indeed, in one outtake, Julia Louis-Dreyfus can’t keep from laughing and spoils take after take. Richards actually gets a little testy with her, he’s so “on.”
Leonard Nimoy was a naturally easy-going, funny, friendly guy and found it very hard to play the repressed, emotionless Spock, but did it suprassingly well.
You’ve never seen a John Wayne or Arnold Schwarzenegger movie, have you?
BoB was pretty remarkable for that- most of the cast was actually British. Damian Lewis (Winters), Ross McCall (Liebgott), Tom Hardy (Janovec), Michael Fassbender (Christensen), and Dexter Fletcher (Martin) are some of the more prominent Brits in the cast.
Another actor who managed to play an American-ish character very well was Jamie Bamber (Apollo) in Battlestar Galactica.
Or Eddie Murphy or Tom Cruise or Lucille Ball or a million other actors who play the same character.
Throw a few more out there …
Ron Perlman as The Beast in Beauty and the Beast
Jaleel White as Steven Urkel in Family Matters
John Hillerman as Higgins in Magnum P.I.
Ian McShane as Al Swearengen in Deadwood
Melissa Rauch as Bernadette on The Big Bang Theory
Debra Jo Rupp as Kitty Forman on That '70s Show.
Megan Mullally as Karen Walker on Will and Grace
(What? You think they talk like that in real life?)
Bob Denver as Maynard and Gilligan and just about everything else. He wasn’t stupid.