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.
The two questions you should ask are:
whether it's the player or the character that's uncomfortable with your proposed shift in mood, and
if it's the latter, whether your players would be OK with the kind of intra-party conflict the shift would generate.
Is the player really OK with it?
If your player is uncomfortable with the campaign turning evil (or at least morally ambiguous) and just "doesn't want to go there", then I'd generally suggest not going there. You started the game with characters that are noble heroes, and all your players seem at least OK with that — you shouldn't change that in the middle of the campaign, unless you're sure your players will be OK with the new theme, too.
If you do still want to try the mood shift, please do make sure to talk things out clearly with the objecting player. Even if they say that it's their character who wouldn't go along with an evil plot, they could be using that as a cover for personal unwillingness to explore such themes. Just make it clear — preferably in private — that they can say "no" if they want, and that you'll respect it.
(Also, no discussion of game themes and intra-group tensions would be complete without a link to the Same Page Tool, so here it is.)
So the player is OK with it, but their character isn't. Now what?
So you have five nominally good characters who are increasingly tempted to "cut corners" and use questionable means to achieve their ends, and one who refuses to do anything of the sort. That's a great recipe for lots of tension between the party members.
This can be a good thing or a bad thing. It can be bad because it requires a tricky balancing act between party cohesion and inter-character conflict, and maintaining that balance can require some skill from both the GM and the players.
On the other hand, it can be good because, done well, it can be really fun to play and makes for great plots.
Note that such conflict doesn't necessarily have to turn into an actual PvP fight (which, based on what I've heard, would generally be a bad idea in 4e anyway). In fact, I'd suggest that, at least 99% of the time, this is something you shouldn't let happen, just like you generally wouldn't let the party get all killed by falling rocks or let the characters just split up and go their separate ways. It just doesn't make for a fun game.
Instead, if you want to try this, you and your players should agree in advance, out of game, to never let things get quite so far that a fight (or a party split) would be unavoidable. This means that the other players — and their characters — need to stay aware of the fact that they need the "pure" character's help (for some reason; preferably a plausible one) and that they therefore have to stop just short of doing anything that this character would find totally unacceptable.
It also means that the player playing that character has to be OK with the fact that their character also, for some reason, needs to cooperate with the others, even if he (the character, that is!) is feeling distinctly uncomfortable about it.
This does require a certain amount of "meta-game" communication between players, and possibly some creative adjustment of character behavior or even plot events to avoid escalating the tension too far. Do tell your players that it's perfectly OK to, say, ask another player out-of-game how their character would react to a particular action by another character, or even to just raise a hand and say "Guys, please don't do that, there's no way my character wouldn't righteously smite you if you did it."
Best Answer
I've run a variety of tones of campaigns over time and some could be considered "evil"; in fact currently I'm running a three-year long Pathfinder campaign where the PCs are pirates - not all of them are technically evilly aligned, but murder, torture, rape, slavery, etc. have all come up in the game. Here's how you make it work.
Decide on Limits, Within Limits
Some people, when they say "evil campaign," just mean "I want to kill lippy villagers like they're orcs," not that they want to really delve into the darker aspects of human nature. Establish an agreement on tone/content with your players up front - you are not required to run (and the players aren't required to participate) in anything they feel like is over their boundaries. I've been known to have players vote on approximate levels of sex, violence, etc. in a game ahead of time, and where they want it to "fade to black."
However, a lot of that will be emergent. In my current pirate campaign, no one really thought about torture until they caught an assassin who was trying to kill the crime-boss they were aligned with. The PC halfling rogue decided he'd torture her extensively to find out who sent her. This definitely put off the other PCs - but not enough that they stopped him. Boundary established (well, lack of one).
Not every "evil" person is 100% evil and on board with everything "evil," though. The ship took two elven women prisoner and one was claimed as a slave by a vicious half-orc pirate. The PC captain didn't really like that but felt somewhat constrained by the expectations of the crew (mutiny is always a threat if the crew doesn't think they're getting their due) so he allowed it. The PCs and that half-orc were having dinner in the captain's cabin, and the halfing from the anecdote above suddenly stabbed the half-orc to death on the dinner table (he's an assassin now - successful death attack). He explained to the shocked command staff that he wouldn't have any slaves on board or associate with slavers. Boundary established.
If you have real characters really roleplaying and thinking through their motivations, you'll still have limits, whether it's "no women, no kids" or the Mafioso that are patriotic and still want neighborhoods to be "family places." Try to depict other "evil" people as complex in that way as well so that they will understand that evil isn't just a race to maximum depravity. With that halfling, torture of captives is OK but slavery and rape is a killin' offense. There's no "Evil Checklist" you have to adhere to and say every crime ever considered is OK - in fact most evil people really are just into one and consider the others to be as bad as other folks do.
However - people make too much of setting boundaries for their games sometimes. If you came up to me and asked me "Do you want to see some chick saw her cheeks off?" I'd say "No! What are you talking about?" But I just went to see the movie Evil Dead, where that exact thing happened as part of the overall horror movie experience. "Boundary pushing" can be good and desirable and allowed based on initial buyin to the general campaign premise. Sure, there's a very slight majority of people so traumatized by something that if it comes up in game it's going to truly trip them out, and there you have outs just like any other kind of media - "press stop," say "I can't deal with this" - but most gaming groups don't really need to do more than establish the general MPAA-rating (e.g. "Hey guys I'm active in my church and I don't really want to go past PG-13 with this game") and then mess around in that area. Other questions here about "oh how do I double check with my players some specific thing is OK before doing it" is overthinking it IMO. If you go see Evil Dead, you'd better expect that if you have a fear of/complex about anything, there's a nonzero chance it's going to come up in lurid color. All the buyin we required for the pirates game was "people can be evil if they want, and expect HBO Original Series level depravity, the pirate world is not a gentle one."
But What's It Really About?
"Evil" is not really a campaign concept (well, not one that passes muster past the 9th grade level). You need a campaign concept and one that will generally keep the PCs acting together instead of being at each others' throats (unless you're looking for a more short, PvP campaign, which is legitimate and there's plenty of short form indie games that facilitate that). If you are more going for "longer D&D campaign," which I'm reading between the lines of your question that you are, it needs to have as much goal as any other campaign. Smart PCs know they need other mighty people to achieve their goals.
Heck most "normal" campaign setups work as well or better with evil groups - just because you're evil, you don't really want where you live and work taken over by zombies or whatever, that interferes with your cashflow. Often times players want to "play evil" because they feel like the GM has been using "goodness" to manipulate them into being passive and they want to be proactive and smart in confronting threats. Squinting too hard at many campaign concepts passed off as "good" reveals them to be a sequence of home invasion, murder, and robbery anyway.
The main trap you're trying to avoid is the PCs just self destructing by going nuts on each other and everyone in the world in general - at least, if they'd be unhappy with being hunted down and slain a couple sessions in.
Actions Have Consequences
Review How do I get my PCs to not be a bunch of murderous cretins? - there are a lot of reasons people don't perform unrestrained evil deeds all the time, from "I don't want to" to "I will get in trouble for it." Sometimes my players complain that the pirate-friendly port city they frequent is "too lawful" just because they can't get away with any heinous crime or breach of the peace they can come up with - but all societies need some kind of stability and will crack down on those affecting that too much. On the other hand, they have become used to not going out into the city alone; traveling in groups is mandatory to not be victimized themselves.
Many evil societies are like this - see how lawful Drow society looks from the outside. Our pirate PCs have to fear their pirates mutinying, the law/navy hunting them down, the bigger pirates in port deciding they're too big for their britches or have so much loot that they're a tempting target in turn. Criminals "hide out" for a reason - they are not free to operate within larger society, and therefore end up having less freedom than good people (something good to play up as the GM). The law, higher level "good" adventurers, etc. are always looking to wipe you out with a clear conscience.
A mechanical option here is keeping track of "infamy points" - I have my own homebrew system I use, but there's a lot of extant reputation-tracking mechanics in the world. People have heard of the big bad people and will react like people do - avoid, confront, narc them out, victimize them, etc. Remember that many victims of crime are doing something bad themselves - criminals, or at least the dishonest, make the best marks for cons and crimes because they have little legal recourse. The pirate PCs can't go just anywhere as their infamy becomes known; honest ports reject them, and other evil folks are generally not the best allies because they like to turn on you when you blink.
Why Do It?
There's a couple reasons to run an evil campaign and the measure of success is different per type.