I tend to do one character at a time, mainly because I don't have the patience to get Mod Organiser to work (it seems like an awesome tool but I couldn't get it to work and it just wound me up...) and I don't want to risk messing up manually changing the active mods in a given game (ADD is a PITA.)
I don't RP super-seriously, in that I sometimes do quests which might not completely fit a given character just because they turn up. Not major questlines, I pick one or two factions and usually one or the other of Dawnguard and Dragonborn (I've only actually completed DG once, and never finished DB, mainly because said ADD tends to lead to restartitis) and I only do the Civil War if I can find a convincing (to me) reason why that character is bothered enough to take a side. Sometimes I do the MQ, sometimes not (have never completed it, either).
But in terms of the smaller quests, if I stumble across something interesting, I'll probably give it a go unless it's *blatantly* wrong for my character - e.g. there's no way Llerethis would have ended up with Mehrunes' Razor, and there's no way Rozenn will be walking around in the Ebony Mail at any point - or with the Ebony Blade, for that matter. She might, conceivably, obtain the Blade through peaceful means and then find a way to dispose of it somewhere safer than its current location, if she happens to hear about strange goings-on, though. I think the Daedric quests are generally among the most RP-friendly (or potentially RP-breaking) quests, because whether you a/ start them b/ continue with them once you figure out what's going on (in some cases, e.g. Molag Bal's quest, arguably Namira's) and c/ complete them this way or that way (Vaermina, Azura, Mehrunes Dagon) can be pretty pivotal moments in a character's 'development', in a way that 'please go and defeat such and such a bandit' doesn't, beyond 'I am some kind of badass who goes around killing bandits'.
I have had a few shortish more seriously RP playthroughs - a huntress who never went anywhere creepier than a bear's cave, a labourer who never strayed off the roads when travelling (but did end up in a few fights to the death because Skyrim loves to throw adventure at you...) and they were great fun, though.
I'm currently playing with Requiem, which I think may help in some ways - it's definitely a long-haul type thing, but I think it'll allow my current character to be rather more realistic than the typical Skyrim mould, because you aren't an amazing killing machine at level 3, there's apparently no way she'll be able to face draugr or anything more monstery than that for a good few levels yet (I've got the difficulty dialed back a bit, but am compensating by not being very good at video games...) so she's starting off wandering around, practicing fighting a bit on animals and being tempted by the idea of bounty hunting, seeing as she appears to have a bit of a knack for fighting with a battleaxe (okay, that's not so realistic, but it *is* a game, I still want it to be fun!) and she's getting pretty good with her flame spell. Eventually, after a much longer period than is usually the case, even when I try to go about things this way, she might feel brave enough to try to go on the frankly ridiculous errand the Jarl's wizard gave her.
Llerethis (my last main character) was pretty badass, but I *did* structure her playthrough to be mostly adventuring around for the first thirty or so levels, with only the very beginning bits of the major questlines I wanted her to do under her belt by that point, and it was much more fun. Unfortunately, I mucked up some mod stuff and ended up abandoning her around level 45, so she never got very far with most of her plans, but previous characters I've got to that level were pretty much done, story wise at that point, and I've abandoned them because they were straight-up overpowered and dull (however much I'd enjoyed playing them up to that point).