that happens best in really well done tv series, where you're allowed a longer time to get to know the characters. plus there are longer and more relatable character arcs. 'the wire' had plenty of that to go around.
in terms of movies... a few that revolve around one single character spring to mind. seymour hoffman's character in 'synecdoche, new york' was really elaborate and consistent, and the movie took place over some 40-50 years, so you really take the journey.
i just mentioned ellen whatsherface's character in 'requiem' in the other thread. that one definitely had me entrenched. i was heartbroken by the end.
this might not qualify under what you wrote in the op, but the lead in 'bronson' gave a really awesome performance and for such an artsy flick, really gave an air of reality to an otherwise visually and narratively exaggerated movie.
larry gopnik from the coen's 'a serious man' has a similar feel to 'synecdoche's caden. successful guys who just don't understand what the f*ck is going on, and do a brilliant job conveying just how clueless they are to the grand scheme taking place around them. i guess the difference between them and almost everybody else is that they actually realize how little they understand.
joel barrish in 'eternal sunshine' as well. oh, and that wrestler from 'the wrestler', randy the rammel rammelstein.
i feel like i'm just listing my favourite recent movies now. i guess the overlap is in the fact that my favourite movies tend to have really well sketched and relatable characters. another one just sprang to mind - tom ripley played by matt damon. damn that movie got into my head.
oh, and one more...