A while back (couple of years ago), I wrote a piece called On Assuming A Character. The link is
here, if you're interested. Essentially, I said that you needed to find a link between who you were and who the character was, and build the character out of that.
But what do you do if you find yourself with a part that's - very much not you? Perhaps you can find common ground, but the character is very different, perhaps even opposing to much of who you are.
At some point, as an actor, it can be pretty much guaranteed that you'll play a part that you're not completely comfortable with. It's a given, and I've certainly played roles that are very different to who I am. My first role was as The Devil, for goodness' sake. (As an aside, it was a great role that I had a lot of fun with, and how I really first got in to acting.) And there's a couple of things that you can do/are often done in this instance.
Firstly, acclimatisation. This is pretty much guaranteed, to be honest. This essentially means making the character a little bit more manageable - trimming off a couple of the rougher edges if you're a bit warmer, adding a bit of bite in if it's a bit too coddly for you. This will pretty much inevitably happen because of who you are, but it can also be a deliberate thing as well.
Secondly, understanding. Often, when you go deep into the script, it will help you to understand the psychology and philosophy of the character, and why they act that way. And sometimes, that will help you to be able to rationalise it a bit (if you're a rationalising sort of person), and find it easier to perform.
Thirdly, adjustment. Not ideal, but at times (particularly if it's a bit more improv-based and such), your character may have a bit of wiggle room, and you can ask the director if you can do things a bit differently. If you can explain it well and show them how your way will still be effective and how you'll be able to do it better that way, they might be willing to give it a go. Not so much for when you've got a set script, and certainly not for bigger stuff.
Fourthly - just do it. There are some bits that you just have to suck it up, swallow your pride, not worry about what your friends or family might think of you for what you're going to do or say, and just go for it. It will be hard the first couple of times, but it gets easier. (In some instances, this is something that you need to watch, because there are some things you don't want to become easy - but that's another discussion.) Don't sacrifice the power and potential of what's written because you're getting a bit queasy. Make the part yours, yes. But you also need to be the part. It's a two-way street; acting isn't (contrary to popular opinion) just about you!