The good opportunities here are massive. It's easy to simply roleplay your way through quests and enjoy the character immersion, but the real benefits can also be had outside of your simple quests. You can (and should!) explore the events surrounding your character by discussing what's going on in-character.
Certainly, I would expect everyone to want to roleplay those stellar, introductory quests in Icecrown. In these quests, you will have the unique opportunity to team up with faction leaders like Lady Sylvanas and Jaina Proudmoore. Not only will you get to team up with them, you'll actually get to fight alongside them. They will treat your character like a hero, as a luminary of your respective faction.
For many of our characters, this will be a final, notable "coming of age." If your character has been struggling as an outcast of your faction, then this could be a final, immutable blessing from a beloved leader. Maybe your character will be surprised and taken off-guard by this level of respect.
More cynical characters should wonder if Jaina is just setting them up. After all, it seems a little foolish to go storming the Lich King's castle with only six people. Maybe Jaina has finally given over to her love for Thrall, and this is a betrayal to trick your crack team of operatives into certain doom.
Sylvanas has got to have her own baggage. You can almost hear the Lament of the Highborn while cruising through the opening quests. Sylvanas takes this fight very personally. We'll talk more about the Forsaken's approach to this conflict a little bit later.
Regardless, the final incursion against the Icecrown Citadel should be treated as a massive, overwhelming thing. After all, this is the final big-to-do of the expansion. This is what (in theory) most of our characters have been working toward for a year. This whole event will live forever in the minds and dreams of Azeroth's races.
Try and remember that your characters are becoming famous. They've just succeeded as the chosen champions from Tirion's trials. Every little nod and twist is could become a character quirk that spells out stories. Do you still distrust Tirion and his plans? Do you travel with a pet? These are the little details that add depth to your roleplay and it's just the kind of detail that the bards of Azeroth will sing about for centuries.
It's not just all war and roses, though. Primordial Saronite is being turned into armor for those storming the Lich King's fortress. Now would be a great time to roleplay your character's doubts and fears about that. (Or, maybe, your character's overwhelming support of such a plan.) Why would it seem like a good idea to clothe your finest warriors in the very blood of an Old God?
The Forsaken still have more than a little explaining to do. It's not like Wrathgate has suddenly become ancient history. During your in-character raids, maybe roleplay through keeping a special eye on the rotting among you. You never know when "Death to the Scourge" is going to make a simple flip over to "Death to the Living." If you are an Undead yourself, now would be a great time to roleplay your vast sorrow for the event -- or maybe your overwhelming indifference.
And don't forget this whole fight is awfully damned personal to the Undead and Death Knights. This fight isn't just about "us versus them" or anything so abstract as "good versus evil." For the Ebon Blade and the Forsaken, Arthas isn't a general enemy. Phrases like "You did this to me" and "I've finally come for you" should be pretty common.
The hardest part about this kind of event, in terms of successful roleplay on a MMORPG, is that you won't be the only person to do those awesome introductory quests. And, in terms of "game reality," you didn't even do them together. How can you reconcile the fact that your 5-man group already killed Scourgelord Tyrannus . . . but so did the guy at the next bar seat over?
There's a couple ways you can reinforce your suspension of disbelief. First, it's pretty easy to assume that it wasn't actually a single five-man group. After all, that was a pretty big assault. Maybe it was dozens of 5-man groups all doing about the same thing. And in the furor of war, you guys just didn't notice each other. Secondly, you can go with the "we didn't finish the job the first time." Maybe they all got better. I know, that's a pretty weak statement, but sometimes you have to stretch disbelief a little thin to make it all work out.
Your last option is to decide that you didn't all fight the same guy. Maybe he fought Tyrannus, and I fought Bobyrannus. This is probably my favorite option for disbelief, because it allows you to place your own, unique spin on the situation. Maybe instead of charging up onto a mountain top for the final boss fight, you found yourself spiraling deeper and deeper into the Pits of Saron. You could also "seed" your own future stories that way. Maybe while there, you saw something frightening that reminded you of old Ulduar, something that you'll become obsessed with for years.
The Real Ugly
The biggest danger zone about roleplaying through the death of Arthas, however, is one of lore. I've seen immense roleplay arguments spring up because folks have different opinion of "what happened before now." It's not even a matter of being right and wrong. World of Warcraft has a massive amount of stories, characters, and events. It's a huge world, and it can be very tough to keep track of it all.
I recommend anyone who's trying to roleplay through the Icecrown Citadel take the time to check out Adam Holisky, Alex Ziebart, and Michael Sacco's collaborative post, The Lore of Patch 3.3. Even if your character has no knowledge of these people or events, it's really best for you, as the player, to have a solid grounding in the shared universe. The story hooks to be found via this lore, of course, are also very helpful when you're trying to plot out your storylines.
Ultimately, this event is going to be about what you can make it. If you just do your quests, the whole thing will fly by, and you'll find yourself sitting in the Cataclysm before you know it. If, however, you take the time to roleplay your character, interact with other people, and explore the lore... you're going to bring yourself a whole deeper level of play.
Good luck in there.