Is it possible to get the best endings playing as a renegade? Or the choices you make (if you are not tempted to break character) are inherently evil at some point, which would prevent them?
Yes. Here is a list of largely spoiler free ending info with requirements listed, you have to click to expand the spoiler for each ending.
Morality doesn't affect whether you can get the "best" ending, though whether or not you saved the Collector base in ME2 forces your hand for certain ending levels.