I've played in and run evil campaigns of various sorts in both 3.5 and 4e (though not 5e, I think my learning will transfer), and run into a lot of problems: My Guy Syndrome comes up a lot, as does a tendency to default to a regular D&D storyline only with more stealing of spoons and kicking of puppies to remind ourselves we're evil. Sometimes an evil campaign instead descends into over-the-top motiveless violence until there's no story at all. There's a whole host of at-the-table and in-the-story issues, and I tried many different strategies to address them. Eventually I came up with a framing device which works well for us in avoiding these problems:
Provide the PCs with a Master to guide them toward orchestrated works of Evil.
Start the game with the PCs as underlings/minions/hirelings/apprentices/etc of a powerful evil NPC. The Master has a complicated Evil Plan and he tasks his minions to enact various parts as the Plan progresses: "Bring me the soul of a hound archon," "Raze the border keep," "Steal the Apocalypse Gem," "Help a spy infiltrate the paladin's ranks," and so forth, tailored to the PCs' abilities.
This provides the party a reason to work together despite having different agendas (and working together will hopefully bond them as friends so that they want to continue as a group) and establishes small achievable evil goals that accumulate into an Epic Evil Event.
All you need to do is ask the players to make sure their characters have a good reason to work for the Master: The serial killer likes having his rampages subsidised (and the Master protects him from the Law); the necromancer seeks to learn from the Master's experience and gain access to his libraries of forbidden lore; the mercenary's in it for the money and benefits.
Eventually the Apprentices will surpass their Master.
Expect the party to betray their Master at some point, hijacking his Evil Plot for their own gain: this is not only expected, but awesome. It's the Master's Evil Plot, not yours, and the story isn't about the Master--it's about his apprentices. Consider the Master to be training wheels for evil, setting an example which the party can then follow to surpass and overthrow their instructor as they level up.
This works because Evil Needs Goals.
As Ed describes so well and AgentPaper elaborates in the D&D context, evil needs concrete reasons motivating its actions. The Master provides goals and motives while the players find their feet in the new paradigm, channeling and guiding their exploration of what it means to be evil in ways compatible with the D&D paradigm without simply kicking puppies during a dungeoncrawl.
A word of warning: Alignment is tricky.
D&D has a history of the details and nature of alignment sparking major heartfelt arguments, because D&D alignments are not easily (or appropriately) matched to real-world philosophies and moralities; they're narrative simplifications to support the game's conceits and draw their power from storytelling conventions rather than from genuine moral complexity. Exactly what this means and how to deal with it are beyond the scope of this answer (and possibly this site, although there's a LOT of questions on the topic you can look at), but you should be aware it exists and be ready to talk with your players about what "Evil campaign" means to them so there aren't nasty surprises mid-game.
"Sentient magic items function as NPCs under the DM's control," says the DMG (214). They have their own personalities; they have Intelligence, Wisdom, and Charisma scores. If they weren't trapped inside swords and boots, they'd be out living their lives and making decisions just like everybody else.
The PHB (122) says celestials and fiends are locked into their alignment, but "for many thinking creatures, alignment is a moral choice." Even the races created by evil gods have free will, and can pursue good alignments if they choose.
I gather that HotDQ isn't very specific about Hazirawn's personality, but if there really is a sentient being in there, you should be able to change its alignment the same way you'd convince a wrathful orc or bloodthirsty gnoll to turn toward Good. It could be a philosophical argument, or a series of object lessons that show the sword the error of its ways. Your chances of success would depend on Hazirawn's own force of personality, which is why I brought up that Charisma score.
Although RAW implies that people can change their alignment, it's up to your DM whether they actually can, and whether an outside agent can impose that kind of change. In any case, my answer is that a sentient magic item, by virtue of possessing a personality, should be treated like a person.
(If you can't hack the philosophical argument angle, you might try to get Hazirawn to draw Balance from the Deck of Many Things...)
Best Answer
Dungeons & Dragons-style Alignment is not cut out for this
The characters in Game of Thrones are almost as complex as real people. Real people cannot be put in one of nine little boxes and call it done. Alignment in general is extremely problematic for a lot of games, but this one especially so. It’s just far too simplistic to handle a “mature” game with any depth. Remember, the system was designed for straight dungeon-delving with little thought for more complicated campaigns. As such, you should not be labeling characters with cute little two-word descriptions like “Lawful Good” or “Chaotic Evil.” You’ll pretty much automatically fail to achieve a complex world full of realistic people with deep motivations and reasons for their actions if you do that.
If you are running this in a system that, like Dungeons & Dragons, has mechanical alignment, you’ll need to handle that. I’ve discussed doing that a few times; this answer sums up the options I see available pretty nicely (and also goes into some more discussion of why alignment is limited).
That said, some thoughts on how those who care about morals can get along with those who don’t, or even those who actively flaunt them for the sake of flaunting them.
Enemy of My Enemy Is My Friend
The classic reason: both Good and Evil face some serious external threat that they must face together. Plenty of Evil has no interest in seeing everything destroyed, or, for that matter, in seeing everything dominated by some other Evil force. Dungeons & Dragons has several examples of these, such as the Blood War (archons, creatures of pure lawful-good-ness, have at times fought alongside the lawful-evil devils against the chaotic-evil demons), and then when the illithids suddenly appeared (which caused the Blood War to stop and potentially had celestials, demons, and devils all fighting together, though that did devolve into bloodshed among the allies).
No One Is Beyond Redemption
The Good guys tolerate the Evil ones on the basis that they can redeem the Evil ones. The Evil ones hang around because the Good guys’ quests (and their rewards) are valuable to the Evil characters. Order of the Stick uses this heavily for Belkar, who is not at all picky about who he cuts up, and the Order provides him with plenty of relatively consequence-free victims. The rest of the group is aware of this but they have some hope (not unfounded, either) of reforming him, or at least preventing him from getting worse.
Evil Can Be Loyal
An Evil character isn’t necessarily a sociopath with no regard for anyone but himself. He may have values and friendships and relationships. He may be Evil because of his methods and his intentions, but that doesn’t necessarily mean he’re a bad guy. This is especially true in the case of simplistic D&D alignment, where you can ding “Evil” without doing anything that strikes a lot of people as horribly evil.