I thoroughly enjoy the podcast Smartless. It's three dudes I like listening to, interviewing people I want to hear from. In the most recent episode I heard, they spoke with comedy legend, Mike Myers. In addition to being a comedic genius, Mike Myers is also a writer and trained actor. He also likes to paint! (Didn't know that, did ya?) In the interview, he said that he only likes to paint people he likes; not people he doesn't. While he has that luxury as a hobbyist, his writing and acting choices need to be more inclusive. For example, Shrek may not be someone Mr. Myers would make godfather to his children, but he needed to connect with the character in order to play him in the movies.
In order to become someone who isn't like him, Myers asks himself "I'm like that when..." So in the case of Shrek, he may have said, "I'm like a reclusive, stinky ogre when... I have the flu and I still have to take out the trash."
When you're writing for a character you're not connecting with, or you're playing a role you don't relate to, you can find common ground with the character by using this wonderful tip from Mike Myers. If you're a compassionate, vegetarian who fights for the rights of all living things, you might struggle to find yourself in writing or performing as a character like Dr. Evil from Austin Powers. But what if you said, "I'm like a psychopathic murderer when... I see people dress up their chihuahuas in demeaning, patronizing costumes for Halloween and I just want to murder them!"
Or what if you're an introverted people-hater who wants nothing to do with group activities of any kind, and you have to write or play a boundary-less cruise director with a penchant for invading people's lives like the Cat in the Hat? You'd probably say, "I'm nothing like this person (cat)." Or you could say, "I'm like a boundary-less cruise director with a penchant for invading people's private lives when... I find out that I've just been gifted a cabin in the woods, miles away from any human being, where I can live for the rest of my life, completely separate from all of mankind." The excitement and elation you feel when you're gifted that hermit-friendly cabin, is the same excitement and elation felt by the person (cat) who wants to instigate disruptive group activites for people they don't know.
In short, don't limit yourself to characters who are "like you". You can find a connection with any person. Or any cat.
Side note: this is also a great exercise for learning to love your (car-idling, leaf-blowing, morally-bankrupt) next-door neighbor.