I've quite happily run one-on-one adventures with no changes from the base rules. A higher focus on minions is handy to come within the XP budget and access to personal healing is a must. It works better with defenders, leaders, and controllers than strikers. (Strikers really do require a party behind them to perform at their best.)
My own experience suggests that you should make it clear that "killing the enemies in open combat" is not the primary tactical motivation, which allows the player to bypass or subvert many encounters.
Instead of multiclassing, potions are an excellent healing mechanism, especially if potions are healing are converted to healing word levels of utility. Alchemy, if made less of a gold-sink, is another excellent way to provide unusual resources to a character.
Well, for starters, I'd say don't use D&D. It is a game tailored towards violent conflicts, which is exactly what you're avoiding, it seems. Mind you, I said "violent conflicts". No story, thus no game, can exist without any conflict whatsoever. I'm not also saying it's completely undoable with D&D, just mainly... a waste of its design and practical goals. Another way to put it, to use a metaphor, is: smartphones are great, you can do a lot with them, they're like handheld computers... But they can't really substitute a desktop computer in every way, maybe not even most ways.
Now, if you're willing to work outside of D&D, there are some good systems out there for that "action and adventure doesn't mean swinging swords all the time" vibe you're after, like, for example, Fate (The Dresden Files RPG, Spirit of the Century, Diaspora, etc), in which most of the mechanics about resolving conflicts are the same, regardless if it's a brawl, a wardrobe and style show off or even an economic dispute between Lex Luthor and Bruce Wayne. It's worth a look, really, and there are SRDs available for some of those games (Spirit of the Century and Diaspora, iirc).
If you're sticking to D&D, plotting the campaign isn't the difficult part; the difficult part is to design encounters (which is just a way of saying "conflict scene") that allow your PCs to shine doing their thing when most of their sheets are geared towards combat (yeah, players do that, it being the game it is).
For the wizard (most hocus-pocus folks, really) and rogue, that's easy. Most other types, though, will rely purely on RP, most of the time, which isn't bad per se, just kinda unfair, since some players get to look at their sheets and say "I can do this, this and that", while the fighter's player has to memorize lines from Gladiator and the ranger's player has to become a living Bear Grylls encyclopedia.
If your group can pull it off, that campaign'd be the stuff of legends, but it'll be hard, really.
Best Answer
Choose 1 of the following: Individualized Plot Decisions, 4th Ed Party Cohesion, Respect Characters Ideals1. At the end of the day, your system and your goals don't match up well.
You can:
At the end of the day, your choices oppose each other and are not particularly compatible with the game system you've chosen. If you want to present choices that matter, they have to matter: PC conflict must be allowed as the losing side must be allowed to escalate (see Dogs in the Vineyard).
While the same page tool is useful here, it's important to begin by articulating your priority of desires. If you want to require group cohesion, set up narrative (and if you can) mechanical incentives for consensus. If you want to present challenging decisions, figure out your characters' motivations and pry at them. You may find that your game of D&D doesn't fulfill your requirements well. This is fine. There are hundreds of game systems out there, and one of them will fit better.
Fourth edition can certainly present morally complicated decisions, but doesn't fufill your requirements as the decisions exist purely in the narrative area of the game. Therefore, they're an excuse to frame different set-piece battles, not have the possibility for party-conflict.
The way to think about it is that 4e is like a big summer blockbuster. There's plot, and there are action sequences. The plot does an absolutely necessary job of tying the action sequences together and making us care about them. In "plot" time, there are very little rules and the protagonists can shout at each other as much as they want... so long as the issues are resolved (to a point where they can banter with each other about them during the fighting at least) by the next "set piece battle."
4e's rules focus on set piece battles between the party and monsters. This conflict is built deeply into the game (mainly due to different rules for monsters and PCs). This means that PCs cannot engage in PvP: there are no rules for it. While you can fake it, fparty-conflict has never worked when I've tried it, save when one person betrays the party, discards their character sheet, and that character becomes a boss-monster for the party to kill.
Note the assumptions there. Things that interact with the party, by definition, die. Good encounter design in 4e is to threaten the party with death without actually killing the party. Note also the protagonist is the party not the individual characters within the party. More so than in all the other D&D games, parties are cohesive groups by the mechanics, and not simply a bunch of fellow travellers.
If you don't want a game full of tactically interesting set piece battles connected by however much plot you're willing to provide and your group is willing to engage in, 4e is not for you. The party, as protagonist, can certainly make these ethically challenging decisions: they inform which battles are to come and what framing goes on in those battles. The PCs can influence the party's decision making process, but cannot defect from it save by defecting from the game.
1On a 10+ choose 2, On a miss, the MC gets to completely derail your campaign.