What purpose does it serve to play not only someone other than you but a person who runs counter to the way you do? More than one, actually. Seeing things from others' shoes bridges understanding. Stepping outside yourself aids in thinking outside the box. Deciding to switch-hit on something you're passionate about can even reinforce your arguments as you see the other side better. There's even risk of changing your opinion on some things. Since there's at least several hundred personality traits I decided to focus just on the "Big Five:" conscientiousness, agreeableness, neuroticism, openness, and extroversion. Figure out where you sit on each of those and pick at least one for a character. Preferably you could pick the one you're the most passionate about or that defines you the most. Now we'll make a character using that concept, only upside-down with it's pants on backwards. Define the character in terms of motivation, aspiration, vehemence, and awareness.
Once you've got the trait(s) you want to portray strongest in this character, it's time to decide how dearly your character regards said trait(s). Go back to how you defined yourself in the beginning. How favorably do you perceive your own level of conscientiousness? Are you passionate about how neurotic or not neurotic you are? Answer this question for yourself first: How much does this aspect of myself drive me? When you've got that down then apply the same to this character. The only difference is where you are passionate one way on the matter they will be passionate the other way.
Something worth identifying is this character's aspiration(s). One more time you'll put yourself in your own shoes to learn about your contradictory character. Is openness valuable to you? Would you like to see everyone else as open as you are? If you have dreams that fall in line with your identified traits then have this character share them as well. Only, make theirs a distorted reflection of yours. They might value privacy highly enough to not only seek it for themselves but impede any system which disrupts it for anyone. I distinguish between motivation and aspiration here because I wish to draw a line between something that a character wants for themselves and something that character wants for everyone. A megalomaniac seeks power but they might not seek to use it against others. Dictatorial megalomaniacs do seek power for the purpose of using it against others. The first is a motivation and the second is an aspiration. While these would be strongly linked, one motivation doesn't necessarily imply any other aspiration. A megalomaniac could spring up in a rebel team where one person just wants to have all the power to protect everyone as they see fit. That same person might have an aspiration to keep everyone getting along and just have power-hungry streaks.
After that it's up to you to decide how strongly the trait(s) manifest themselves, as well as how often. Frequently or strongly displayed traits are easy to spot. That doesn't mean they'd have to be aware of them, though. Also note that just because something drives someone doesn't mean it's always present; Differences in motivation from personality happens when we reach a crossroads more often than on straightaways. Is this character very non-conscientious? Maybe they act in accordance with their conscience only when others are watching. Perhaps if they're given the opportunity they'd steal only some of the party treasure, or perhaps they'd take it all.
Next consider how close that strong trait is to their mind. How often to they realize they exhibit it? I know plenty of stubborn people who don't always realize they are being stubborn. If you're always friendly to new people then this character might always be more interested in their own thoughts. That would constitute at least major awareness. Do people tell you that you worry too much but, even though you're inclined to believe them, you don't see it? Concoct someone who is oblivious to the fact that they are oblivious to all kinds of threats, or the level of danger which could ensue from them. Mirror yourself again. Someone not very aware of their own lack (or surplus) of extroversion can still be driven by it – they just perceive it as something else.
Motivation and aspiration are the internal factors for your character's personality. While my friends and family should know my motivations by now and at least have a good idea of my aspirations, the person I just met at the supermarket wouldn't have as good a picture of me. The strength and frequency of their defining traits are the external factors. Very, very closed off people are usually pretty obvious – just say more than a sentence to them. All the mitigation between the internal and externally visible personality falls upon the character's awareness of that trait. My awareness of the fact that I'm very agreeable means I can better choose if I should or should not be agreeable given circumstances, but only for as long as I remain aware of it. In that same way the very introverted character could make an effort to not be so introverted and thus make that trait less apparent to others. Happy contradicting!